Precalculus Examples

$\log (8 (\frac{25 (29 + 5 \sqrt{33})}{8})) - \log (1 + \sqrt{\frac{25 (29 + 5 \sqrt{33})}{8}})$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{8 (\frac{25 (29 + 5 \sqrt{33})}{8})}{1 + \sqrt{\frac{25 (29 + 5 \sqrt{33})}{8}}})$

Step 2

Combine $8$ and $\frac{25 (29 + 5 \sqrt{33})}{8}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \sqrt{\frac{25 (29 + 5 \sqrt{33})}{8}}})$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt{\frac{25 (29 + 5 \sqrt{33})}{8}}$ as $\frac{\sqrt{25 (29 + 5 \sqrt{33})}}{\sqrt{8}}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{\sqrt{25 (29 + 5 \sqrt{33})}}{\sqrt{8}}})$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{\sqrt{5^{2} (29 + 5 \sqrt{33})}}{\sqrt{8}}})$

Step 3.2.2

Pull terms out from under the radical.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}}}{\sqrt{8}}})$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.3.1.1

Factor $4$ out of $8$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}}}{\sqrt{4 (2)}}})$

Step 3.3.1.2

Rewrite $4$ as $2^{2}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}}}{\sqrt{2^{2} \cdot 2}}})$

Step 3.3.2

Pull terms out from under the radical.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}}}{2 \sqrt{2}}})$

Step 3.4

Multiply $\frac{5 \sqrt{29 + 5 \sqrt{33}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{5 \sqrt{29 + 5 \sqrt{33}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}})$

Step 3.5.2

Move $\sqrt{2}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}})$

Step 3.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}})$

Step 3.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}})$

Step 3.5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}})$

Step 3.5.6

Add $1$ and $1$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 {\sqrt{2}}^{2}}})$

Step 3.5.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}})$

Step 3.5.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}})$

Step 3.5.7.3

Combine $\frac{1}{2}$ and $2$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}})$

Step 3.5.7.4

Cancel the common factor of $2$ .

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Step 3.5.7.4.1

Cancel the common factor.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}})$

Step 3.5.7.4.2

Rewrite the expression.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \cdot 2^{1}}})$

Step 3.5.7.5

Evaluate the exponent.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{29 + 5 \sqrt{33}} \sqrt{2}}{2 \cdot 2}})$

Step 3.6

Combine using the product rule for radicals.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{2 (29 + 5 \sqrt{33})}}{2 \cdot 2}})$

Step 3.7

Multiply $2$ by $2$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{2 (29 + 5 \sqrt{33})}}{4}})$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Apply the distributive property.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{2 \cdot 29 + 2 (5 \sqrt{33})}}{4}})$

Step 3.8.2

Multiply $2$ by $29$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{58 + 2 (5 \sqrt{33})}}{4}})$

Step 3.8.3

Multiply $5$ by $2$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{1 + \frac{5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 3.9

Write $1$ as a fraction with a common denominator.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{\frac{4}{4} + \frac{5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 3.10

Combine the numerators over the common denominator.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 4

Multiply $8$ by $25$ .

$\log (\frac{\frac{200 (29 + 5 \sqrt{33})}{8}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Reduce the expression $\frac{200 (29 + 5 \sqrt{33})}{8}$ by cancelling the common factors.

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Step 5.1.1

Factor $8$ out of $200 (29 + 5 \sqrt{33})$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 5.1.2

Factor $8$ out of $8$ .

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8 (1)}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 5.1.3

Cancel the common factor.

$\log (\frac{\frac{8 (25 (29 + 5 \sqrt{33}))}{8 \cdot 1}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 5.1.4

Rewrite the expression.

$\log (\frac{\frac{25 (29 + 5 \sqrt{33})}{1}}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 5.2

Divide $25 (29 + 5 \sqrt{33})$ by $1$ .

$\log (\frac{25 (29 + 5 \sqrt{33})}{\frac{4 + 5 \sqrt{58 + 10 \sqrt{33}}}{4}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\log (25 (29 + 5 \sqrt{33}) \frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}})$

Step 7

Apply the distributive property.

$\log ((25 \cdot 29 + 25 (5 \sqrt{33})) \frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}})$

Step 8

Multiply.

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Step 8.1

Multiply $25$ by $29$ .

$\log ((725 + 25 (5 \sqrt{33})) \frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}})$

Step 8.2

Multiply $5$ by $25$ .

$\log ((725 + 125 \sqrt{33}) \frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}})$

Step 9

Multiply $\frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}}$ by $\frac{4 - 5 \sqrt{58 + 10 \sqrt{33}}}{4 - 5 \sqrt{58 + 10 \sqrt{33}}}$ .

$\log ((725 + 125 \sqrt{33}) (\frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}} \cdot \frac{4 - 5 \sqrt{58 + 10 \sqrt{33}}}{4 - 5 \sqrt{58 + 10 \sqrt{33}}}))$

Step 10

Multiply $\frac{4}{4 + 5 \sqrt{58 + 10 \sqrt{33}}}$ by $\frac{4 - 5 \sqrt{58 + 10 \sqrt{33}}}{4 - 5 \sqrt{58 + 10 \sqrt{33}}}$ .

$\log ((725 + 125 \sqrt{33}) \frac{4 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{(4 + 5 \sqrt{58 + 10 \sqrt{33}}) (4 - 5 \sqrt{58 + 10 \sqrt{33}})})$

Step 11

Expand the denominator using the FOIL method.

$\log ((725 + 125 \sqrt{33}) \frac{4 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{16 - 20 \sqrt{58 + 10 \sqrt{33}} + 20 \sqrt{58 + 10 \sqrt{33}} - 25 {\sqrt{58 + 10 \sqrt{33}}}^{2}})$

Step 12

Simplify.

$\log ((725 + 125 \sqrt{33}) \frac{4 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{- 1434 - 250 \sqrt{33}})$

Step 13

Cancel the common factor of $4$ and $- 1434 - 250 \sqrt{33}$ .

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Step 13.1

Factor $2$ out of $4 (4 - 5 \sqrt{58 + 10 \sqrt{33}})$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 (2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}))}{- 1434 - 250 \sqrt{33}})$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $2$ out of $- 1434$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 (2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}))}{2 (- 717) - 250 \sqrt{33}})$

Step 13.2.2

Factor $2$ out of $- 250 \sqrt{33}$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 (2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}))}{2 (- 717) + 2 (- 125 \sqrt{33})})$

Step 13.2.3

Factor $2$ out of $2 (- 717) + 2 (- 125 \sqrt{33})$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 (2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}))}{2 (- 717 - 125 \sqrt{33})})$

Step 13.2.4

Cancel the common factor.

$\log ((725 + 125 \sqrt{33}) \frac{2 (2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}))}{2 (- 717 - 125 \sqrt{33})})$

Step 13.2.5

Rewrite the expression.

$\log ((725 + 125 \sqrt{33}) \frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{- 717 - 125 \sqrt{33}})$

Step 14

Multiply $\frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{- 717 - 125 \sqrt{33}}$ by $\frac{- 717 + 125 \sqrt{33}}{- 717 + 125 \sqrt{33}}$ .

$\log ((725 + 125 \sqrt{33}) (\frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{- 717 - 125 \sqrt{33}} \cdot \frac{- 717 + 125 \sqrt{33}}{- 717 + 125 \sqrt{33}}))$

Step 15

Simplify terms.

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Step 15.1

Multiply $\frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}})}{- 717 - 125 \sqrt{33}}$ by $\frac{- 717 + 125 \sqrt{33}}{- 717 + 125 \sqrt{33}}$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})}{(- 717 - 125 \sqrt{33}) (- 717 + 125 \sqrt{33})})$

Step 15.2

Expand the denominator using the FOIL method.

$\log ((725 + 125 \sqrt{33}) \frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})}{514089 - 89625 \sqrt{33} + 89625 \sqrt{33} - 15625 {\sqrt{33}}^{2}})$

Step 15.3

Simplify.

$\log ((725 + 125 \sqrt{33}) \frac{2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})}{- 1536})$

Step 15.4

Cancel the common factor of $2$ and $- 1536$ .

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Step 15.4.1

Factor $2$ out of $2 (4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 ((4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33}))}{- 1536})$

Step 15.4.2

Cancel the common factors.

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Step 15.4.2.1

Factor $2$ out of $- 1536$ .

$\log ((725 + 125 \sqrt{33}) \frac{2 ((4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33}))}{2 \cdot - 768})$

Step 15.4.2.2

Cancel the common factor.

$\log ((725 + 125 \sqrt{33}) \frac{2 ((4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33}))}{2 \cdot - 768})$

Step 15.4.2.3

Rewrite the expression.

$\log ((725 + 125 \sqrt{33}) \frac{(4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})}{- 768})$

Step 16

Expand $(4 - 5 \sqrt{58 + 10 \sqrt{33}}) (- 717 + 125 \sqrt{33})$ using the FOIL Method.

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Step 16.1

Apply the distributive property.

$\log ((725 + 125 \sqrt{33}) \frac{4 (- 717 + 125 \sqrt{33}) - 5 \sqrt{58 + 10 \sqrt{33}} (- 717 + 125 \sqrt{33})}{- 768})$

Step 16.2

Apply the distributive property.

$\log ((725 + 125 \sqrt{33}) \frac{4 \cdot - 717 + 4 (125 \sqrt{33}) - 5 \sqrt{58 + 10 \sqrt{33}} (- 717 + 125 \sqrt{33})}{- 768})$

Step 16.3

Apply the distributive property.

$\log ((725 + 125 \sqrt{33}) \frac{4 \cdot - 717 + 4 (125 \sqrt{33}) - 5 \sqrt{58 + 10 \sqrt{33}} \cdot - 717 - 5 \sqrt{58 + 10 \sqrt{33}} (125 \sqrt{33})}{- 768})$

Step 17

Simplify terms.

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Step 17.1

Simplify each term.

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Step 17.1.1

Multiply $4$ by $- 717$ .

$\log ((725 + 125 \sqrt{33}) \frac{- 2868 + 4 (125 \sqrt{33}) - 5 \sqrt{58 + 10 \sqrt{33}} \cdot - 717 - 5 \sqrt{58 + 10 \sqrt{33}} (125 \sqrt{33})}{- 768})$

Step 17.1.2

Multiply $125$ by $4$ .

$\log ((725 + 125 \sqrt{33}) \frac{- 2868 + 500 \sqrt{33} - 5 \sqrt{58 + 10 \sqrt{33}} \cdot - 717 - 5 \sqrt{58 + 10 \sqrt{33}} (125 \sqrt{33})}{- 768})$

Step 17.1.3

Multiply $- 717$ by $- 5$ .

$\log ((725 + 125 \sqrt{33}) \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 5 \sqrt{58 + 10 \sqrt{33}} (125 \sqrt{33})}{- 768})$

Step 17.1.4

Multiply $- 5 \sqrt{58 + 10 \sqrt{33}} (125 \sqrt{33})$ .

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Step 17.1.4.1

Multiply $125$ by $- 5$ .

$\log ((725 + 125 \sqrt{33}) \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{58 + 10 \sqrt{33}} \sqrt{33}}{- 768})$

Step 17.1.4.2

Combine using the product rule for radicals.

$\log ((725 + 125 \sqrt{33}) \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{- 768})$

Step 17.2

Simplify by multiplying through.

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Step 17.2.1

Move the negative in front of the fraction.

$\log ((725 + 125 \sqrt{33}) (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}))$

Step 17.2.2

Apply the distributive property.

$\log (725 (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}) + 125 \sqrt{33} (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}))$

Step 18

Multiply $725 (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$ .

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Step 18.1

Multiply $- 1$ by $725$ .

$\log (- 725 \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768} + 125 \sqrt{33} (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}))$

Step 18.2

Combine $- 725$ and $\frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}$ .

$\log (\frac{- 725 (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})})}{768} + 125 \sqrt{33} (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}))$

Step 19

Multiply $125 \sqrt{33} (- \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$ .

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Step 19.1

Multiply $- 1$ by $125$ .

$\log (\frac{- 725 (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})})}{768} - 125 \sqrt{33} \frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 19.2

Combine $\frac{- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}}{768}$ and $- 125$ .

$\log (\frac{- 725 (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})})}{768} + \frac{(- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125}{768} \sqrt{33})$

Step 19.3

Combine $\frac{(- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125}{768}$ and $\sqrt{33}$ .

Step 20

Combine the numerators over the common denominator.

$\log (\frac{- 725 (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21

Simplify each term.

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Step 21.1

Apply the distributive property.

$\log (\frac{- 725 \cdot - 2868 - 725 (500 \sqrt{33}) - 725 (3585 \sqrt{58 + 10 \sqrt{33}}) - 725 (- 625 \sqrt{33 (58 + 10 \sqrt{33})}) + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21.2

Simplify.

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Step 21.2.1

Multiply $- 725$ by $- 2868$ .

$\log (\frac{2079300 - 725 (500 \sqrt{33}) - 725 (3585 \sqrt{58 + 10 \sqrt{33}}) - 725 (- 625 \sqrt{33 (58 + 10 \sqrt{33})}) + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21.2.2

Multiply $500$ by $- 725$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 725 (3585 \sqrt{58 + 10 \sqrt{33}}) - 725 (- 625 \sqrt{33 (58 + 10 \sqrt{33})}) + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21.2.3

Multiply $3585$ by $- 725$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} - 725 (- 625 \sqrt{33 (58 + 10 \sqrt{33})}) + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21.2.4

Multiply $- 625$ by $- 725$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (- 2868 + 500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})}) \cdot - 125 \sqrt{33}}{768})$

Step 21.3

Apply the distributive property.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (- 2868 \cdot - 125 + 500 \sqrt{33} \cdot - 125 + 3585 \sqrt{58 + 10 \sqrt{33}} \cdot - 125 - 625 \sqrt{33 (58 + 10 \sqrt{33})} \cdot - 125) \sqrt{33}}{768})$

Step 21.4

Simplify.

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Step 21.4.1

Multiply $- 2868$ by $- 125$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (358500 + 500 \sqrt{33} \cdot - 125 + 3585 \sqrt{58 + 10 \sqrt{33}} \cdot - 125 - 625 \sqrt{33 (58 + 10 \sqrt{33})} \cdot - 125) \sqrt{33}}{768})$

Step 21.4.2

Multiply $- 125$ by $500$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (358500 - 62500 \sqrt{33} + 3585 \sqrt{58 + 10 \sqrt{33}} \cdot - 125 - 625 \sqrt{33 (58 + 10 \sqrt{33})} \cdot - 125) \sqrt{33}}{768})$

Step 21.4.3

Multiply $- 125$ by $3585$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (358500 - 62500 \sqrt{33} - 448125 \sqrt{58 + 10 \sqrt{33}} - 625 \sqrt{33 (58 + 10 \sqrt{33})} \cdot - 125) \sqrt{33}}{768})$

Step 21.4.4

Multiply $- 125$ by $- 625$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + (358500 - 62500 \sqrt{33} - 448125 \sqrt{58 + 10 \sqrt{33}} + 78125 \sqrt{33 (58 + 10 \sqrt{33})}) \sqrt{33}}{768})$

Step 21.5

Apply the distributive property.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 \sqrt{33} \sqrt{33} - 448125 \sqrt{58 + 10 \sqrt{33}} \sqrt{33} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6

Simplify.

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Step 21.6.1

Multiply $- 62500 \sqrt{33} \sqrt{33}$ .

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Step 21.6.1.1

Raise $\sqrt{33}$ to the power of $1$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 ({\sqrt{33}}^{1} \sqrt{33}) - 448125 \sqrt{58 + 10 \sqrt{33}} \sqrt{33} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6.1.2

Raise $\sqrt{33}$ to the power of $1$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 ({\sqrt{33}}^{1} {\sqrt{33}}^{1}) - 448125 \sqrt{58 + 10 \sqrt{33}} \sqrt{33} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 {\sqrt{33}}^{1 + 1} - 448125 \sqrt{58 + 10 \sqrt{33}} \sqrt{33} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6.1.4

Add $1$ and $1$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 {\sqrt{33}}^{2} - 448125 \sqrt{58 + 10 \sqrt{33}} \sqrt{33} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6.2

Combine using the product rule for radicals.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 {\sqrt{33}}^{2} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}}{768})$

Step 21.6.3

Multiply $78125 \sqrt{33 (58 + 10 \sqrt{33})} \sqrt{33}$ .

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Step 21.6.3.1

Combine using the product rule for radicals.

Step 21.6.3.2

Multiply $33$ by $33$ .

Step 21.7

Simplify each term.

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Step 21.7.1

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 21.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 {(33^{\frac{1}{2}})}^{2} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 \cdot 33^{\frac{1}{2} \cdot 2} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.1.3

Combine $\frac{1}{2}$ and $2$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 \cdot 33^{\frac{2}{2}} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.1.4

Cancel the common factor of $2$ .

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Step 21.7.1.4.1

Cancel the common factor.

Step 21.7.1.4.2

Rewrite the expression.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 \cdot 33^{1} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.1.5

Evaluate the exponent.

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 62500 \cdot 33 - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.2

Multiply $- 62500$ by $33$ .

$\log (\frac{2079300 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 2062500 - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 78125 \sqrt{1089 (58 + 10 \sqrt{33})}}{768})$

Step 21.7.3

Rewrite $1089$ as $33^{2}$ .

Step 21.7.4

Pull terms out from under the radical.

Step 21.7.5

Multiply $33$ by $78125$ .

Step 22

Simplify terms.

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Step 22.1

Subtract $2062500$ from $2079300$ .

$\log (\frac{16800 - 362500 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} + 358500 \sqrt{33} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 2578125 \sqrt{58 + 10 \sqrt{33}}}{768})$

Step 22.2

Add $- 362500 \sqrt{33}$ and $358500 \sqrt{33}$ .

$\log (\frac{16800 - 4000 \sqrt{33} - 2599125 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} - 448125 \sqrt{33 (58 + 10 \sqrt{33})} + 2578125 \sqrt{58 + 10 \sqrt{33}}}{768})$

Step 22.3

Add $- 2599125 \sqrt{58 + 10 \sqrt{33}}$ and $2578125 \sqrt{58 + 10 \sqrt{33}}$ .

$\log (\frac{16800 - 4000 \sqrt{33} - 21000 \sqrt{58 + 10 \sqrt{33}} + 453125 \sqrt{33 (58 + 10 \sqrt{33})} - 448125 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.4

Subtract $448125 \sqrt{33 (58 + 10 \sqrt{33})}$ from $453125 \sqrt{33 (58 + 10 \sqrt{33})}$ .

$\log (\frac{16800 - 4000 \sqrt{33} - 21000 \sqrt{58 + 10 \sqrt{33}} + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5

Cancel the common factor of $16800 - 4000 \sqrt{33} - 21000 \sqrt{58 + 10 \sqrt{33}} + 5000 \sqrt{33 (58 + 10 \sqrt{33})}$ and $768$ .

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Step 22.5.1

Factor $8$ out of $16800$ .

$\log (\frac{8 (2100) - 4000 \sqrt{33} - 21000 \sqrt{58 + 10 \sqrt{33}} + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5.2

Factor $8$ out of $- 4000 \sqrt{33}$ .

$\log (\frac{8 (2100) + 8 (- 500 \sqrt{33}) - 21000 \sqrt{58 + 10 \sqrt{33}} + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5.3

Factor $8$ out of $8 (2100) + 8 (- 500 \sqrt{33})$ .

$\log (\frac{8 (2100 - 500 \sqrt{33}) - 21000 \sqrt{58 + 10 \sqrt{33}} + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5.4

Factor $8$ out of $- 21000 \sqrt{58 + 10 \sqrt{33}}$ .

$\log (\frac{8 (2100 - 500 \sqrt{33}) + 8 (- 2625 \sqrt{58 + 10 \sqrt{33}}) + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5.5

Factor $8$ out of $8 (2100 - 500 \sqrt{33}) + 8 (- 2625 \sqrt{58 + 10 \sqrt{33}})$ .

$\log (\frac{8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}}) + 5000 \sqrt{33 (58 + 10 \sqrt{33})}}{768})$

Step 22.5.6

Factor $8$ out of $5000 \sqrt{33 (58 + 10 \sqrt{33})}$ .

$\log (\frac{8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}}) + 8 (625 \sqrt{33 (58 + 10 \sqrt{33})})}{768})$

Step 22.5.7

Factor $8$ out of $8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}}) + 8 (625 \sqrt{33 (58 + 10 \sqrt{33})})$ .

$\log (\frac{8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}} + 625 \sqrt{33 (58 + 10 \sqrt{33})})}{768})$

Step 22.5.8

Cancel the common factors.

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Step 22.5.8.1

Factor $8$ out of $768$ .

$\log (\frac{8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}} + 625 \sqrt{33 (58 + 10 \sqrt{33})})}{8 \cdot 96})$

Step 22.5.8.2

Cancel the common factor.

$\log (\frac{8 (2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}} + 625 \sqrt{33 (58 + 10 \sqrt{33})})}{8 \cdot 96})$

Step 22.5.8.3

Rewrite the expression.

$\log (\frac{2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}} + 625 \sqrt{33 (58 + 10 \sqrt{33})}}{96})$

Step 23

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{2100 - 500 \sqrt{33} - 2625 \sqrt{58 + 10 \sqrt{33}} + 625 \sqrt{33 (58 + 10 \sqrt{33})}}{96})$

Decimal Form:

$2$