Trigonometry Examples

$\log_{8} (5 x) - \log_{8} (13 + \sqrt{x}) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{8} (\frac{5 x}{13 + \sqrt{x}}) = 2$

Step 1.2

Multiply $\frac{5 x}{13 + \sqrt{x}}$ by $\frac{13 - \sqrt{x}}{13 - \sqrt{x}}$ .

$\log_{8} (\frac{5 x}{13 + \sqrt{x}} \cdot \frac{13 - \sqrt{x}}{13 - \sqrt{x}}) = 2$

Step 1.3

Multiply $\frac{5 x}{13 + \sqrt{x}}$ by $\frac{13 - \sqrt{x}}{13 - \sqrt{x}}$ .

$\log_{8} (\frac{5 x (13 - \sqrt{x})}{(13 + \sqrt{x}) (13 - \sqrt{x})}) = 2$

Step 1.4

Expand the denominator using the FOIL method.

$\log_{8} (\frac{5 x (13 - \sqrt{x})}{169 - 13 \sqrt{x} + \sqrt{x} \cdot 13 - {\sqrt{x}}^{2}}) = 2$

Step 1.5

Simplify.

$\log_{8} (\frac{5 x (13 - \sqrt{x})}{- x + 169}) = 2$

Step 2

Rewrite $\log_{8} (\frac{5 x (13 - \sqrt{x})}{- x + 169}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$8^{2} = \frac{5 x (13 - \sqrt{x})}{- x + 169}$

Step 3

Cross multiply to remove the fraction.

$5 x (13 - \sqrt{x}) = 8^{2} (- x + 169)$

Step 4

Simplify $8^{2} (- x + 169)$ .

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Step 4.1

Raise $8$ to the power of $2$ .

$5 x (13 - \sqrt{x}) = 64 (- x + 169)$

Step 4.2

Apply the distributive property.

$5 x (13 - \sqrt{x}) = 64 (- x) + 64 \cdot 169$

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $64$ .

$5 x (13 - \sqrt{x}) = - 64 x + 64 \cdot 169$

Step 4.3.2

Multiply $64$ by $169$ .

$5 x (13 - \sqrt{x}) = - 64 x + 10816$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Add $64 x$ to both sides of the equation.

$5 x (13 - \sqrt{x}) + 64 x = 10816$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$5 x \cdot 13 + 5 x (- \sqrt{x}) + 64 x = 10816$

Step 5.2.2

Multiply $13$ by $5$ .

$65 x + 5 x (- \sqrt{x}) + 64 x = 10816$

Step 5.2.3

Multiply $- 1$ by $5$ .

$65 x - 5 x \sqrt{x} + 64 x = 10816$

Step 5.3

Add $65 x$ and $64 x$ .

$- 5 x \sqrt{x} + 129 x = 10816$

Step 6

Factor $x$ out of $- 5 x \sqrt{x} + 129 x$ .

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Step 6.1

Factor $x$ out of $- 5 x \sqrt{x}$ .

$x (- 5 \sqrt{x}) + 129 x = 10816$

Step 6.2

Factor $x$ out of $129 x$ .

$x (- 5 \sqrt{x}) + x \cdot 129 = 10816$

Step 6.3

Factor $x$ out of $x (- 5 \sqrt{x}) + x \cdot 129$ .

$x (- 5 \sqrt{x} + 129) = 10816$

Step 7

Solve for $- 5 \sqrt{x}$ .

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Step 7.1

Divide each term in $x (- 5 \sqrt{x} + 129) = 10816$ by $x$ and simplify.

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Step 7.1.1

Divide each term in $x (- 5 \sqrt{x} + 129) = 10816$ by $x$ .

$\frac{x (- 5 \sqrt{x} + 129)}{x} = \frac{10816}{x}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Cancel the common factor of $x$ .

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Step 7.1.2.1.1

Cancel the common factor.

$\frac{x (- 5 \sqrt{x} + 129)}{x} = \frac{10816}{x}$

Step 7.1.2.1.2

Divide $- 5 \sqrt{x} + 129$ by $1$ .

$- 5 \sqrt{x} + 129 = \frac{10816}{x}$

Step 7.2

Subtract $129$ from both sides of the equation.

$- 5 \sqrt{x} = \frac{10816}{x} - 129$

Step 8

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 5 \sqrt{x})}^{2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9

Simplify each side of the equation.

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Step 9.1

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

${(- 5 x^{\frac{1}{2}})}^{2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Simplify ${(- 5 x^{\frac{1}{2}})}^{2}$ .

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Step 9.2.1.1

Apply the product rule to $- 5 x^{\frac{1}{2}}$ .

${(- 5)}^{2} {(x^{\frac{1}{2}})}^{2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2.1.2

Raise $- 5$ to the power of $2$ .

$25 {(x^{\frac{1}{2}})}^{2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 9.2.1.3.1

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

$25 x^{\frac{1}{2} \cdot 2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2.1.3.2

Cancel the common factor of $2$ .

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Step 9.2.1.3.2.1

Cancel the common factor.

$25 x^{\frac{1}{2} \cdot 2} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2.1.3.2.2

Rewrite the expression.

$25 x^{1} = {(\frac{10816}{x} - 129)}^{2}$

Step 9.2.1.4

Simplify.

$25 x = {(\frac{10816}{x} - 129)}^{2}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify ${(\frac{10816}{x} - 129)}^{2}$ .

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Step 9.3.1.1

Rewrite ${(\frac{10816}{x} - 129)}^{2}$ as $(\frac{10816}{x} - 129) (\frac{10816}{x} - 129)$ .

$25 x = (\frac{10816}{x} - 129) (\frac{10816}{x} - 129)$

Step 9.3.1.2

Expand $(\frac{10816}{x} - 129) (\frac{10816}{x} - 129)$ using the FOIL Method.

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Step 9.3.1.2.1

Apply the distributive property.

$25 x = \frac{10816}{x} (\frac{10816}{x} - 129) - 129 (\frac{10816}{x} - 129)$

Step 9.3.1.2.2

Apply the distributive property.

$25 x = \frac{10816}{x} \cdot \frac{10816}{x} + \frac{10816}{x} \cdot - 129 - 129 (\frac{10816}{x} - 129)$

Step 9.3.1.2.3

Apply the distributive property.

$25 x = \frac{10816}{x} \cdot \frac{10816}{x} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3

Simplify and combine like terms.

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Step 9.3.1.3.1

Simplify each term.

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Step 9.3.1.3.1.1

Multiply $\frac{10816}{x} \cdot \frac{10816}{x}$ .

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Step 9.3.1.3.1.1.1

Multiply $\frac{10816}{x}$ by $\frac{10816}{x}$ .

$25 x = \frac{10816 \cdot 10816}{x \cdot x} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.1.2

Multiply $10816$ by $10816$ .

$25 x = \frac{116985856}{x \cdot x} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.1.3

Raise $x$ to the power of $1$ .

$25 x = \frac{116985856}{x^{1} x} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.1.4

Raise $x$ to the power of $1$ .

$25 x = \frac{116985856}{x^{1} x^{1}} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.1.5

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

$25 x = \frac{116985856}{x^{1 + 1}} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.1.6

Add $1$ and $1$ .

$25 x = \frac{116985856}{x^{2}} + \frac{10816}{x} \cdot - 129 - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.2

Multiply $\frac{10816}{x} \cdot - 129$ .

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Step 9.3.1.3.1.2.1

Combine $\frac{10816}{x}$ and $- 129$ .

$25 x = \frac{116985856}{x^{2}} + \frac{10816 \cdot - 129}{x} - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.2.2

Multiply $10816$ by $- 129$ .

$25 x = \frac{116985856}{x^{2}} + \frac{- 1395264}{x} - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.3

Move the negative in front of the fraction.

$25 x = \frac{116985856}{x^{2}} - \frac{1395264}{x} - 129 \frac{10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.4

Multiply $- 129 \frac{10816}{x}$ .

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Step 9.3.1.3.1.4.1

Combine $- 129$ and $\frac{10816}{x}$ .

$25 x = \frac{116985856}{x^{2}} - \frac{1395264}{x} + \frac{- 129 \cdot 10816}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.4.2

Multiply $- 129$ by $10816$ .

$25 x = \frac{116985856}{x^{2}} - \frac{1395264}{x} + \frac{- 1395264}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.5

Move the negative in front of the fraction.

$25 x = \frac{116985856}{x^{2}} - \frac{1395264}{x} - \frac{1395264}{x} - 129 \cdot - 129$

Step 9.3.1.3.1.6

Multiply $- 129$ by $- 129$ .

$25 x = \frac{116985856}{x^{2}} - \frac{1395264}{x} - \frac{1395264}{x} + 16641$

Step 9.3.1.3.2

Subtract $\frac{1395264}{x}$ from $- \frac{1395264}{x}$ .

$25 x = \frac{116985856}{x^{2}} - 2 \frac{1395264}{x} + 16641$

Step 9.3.1.4

Simplify each term.

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Step 9.3.1.4.1

Multiply $- 2 \frac{1395264}{x}$ .

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Step 9.3.1.4.1.1

Combine $- 2$ and $\frac{1395264}{x}$ .

$25 x = \frac{116985856}{x^{2}} + \frac{- 2 \cdot 1395264}{x} + 16641$

Step 9.3.1.4.1.2

Multiply $- 2$ by $1395264$ .

$25 x = \frac{116985856}{x^{2}} + \frac{- 2790528}{x} + 16641$

Step 9.3.1.4.2

Move the negative in front of the fraction.

$25 x = \frac{116985856}{x^{2}} - \frac{2790528}{x} + 16641$

Step 10

Solve for $x$ .

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Step 10.1

Find the LCD of the terms in the equation.

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Step 10.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, x, 1$

Step 10.1.2

Since $1, x^{2}, x, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{2}, x^{1}$ .

Step 10.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 10.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 10.1.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 10.1.6

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 10.1.7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 10.1.8

The LCM of $x^{2}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 10.1.9

Multiply $x$ by $x$ .

$x^{2}$

Step 10.2

Multiply each term in $25 x = \frac{116985856}{x^{2}} - \frac{2790528}{x} + 16641$ by $x^{2}$ to eliminate the fractions.

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Step 10.2.1

Multiply each term in $25 x = \frac{116985856}{x^{2}} - \frac{2790528}{x} + 16641$ by $x^{2}$ .

$25 x \cdot x^{2} = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 10.2.2.1.1

Move $x^{2}$ .

$25 (x^{2} x) = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.2.1.2

Multiply $x^{2}$ by $x$ .

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Step 10.2.2.1.2.1

Raise $x$ to the power of $1$ .

$25 (x^{2} x^{1}) = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.2.1.2.2

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

$25 x^{2 + 1} = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.2.1.3

Add $2$ and $1$ .

$25 x^{3} = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Simplify each term.

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Step 10.2.3.1.1

Cancel the common factor of $x^{2}$ .

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Step 10.2.3.1.1.1

Cancel the common factor.

$25 x^{3} = \frac{116985856}{x^{2}} x^{2} - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.3.1.1.2

Rewrite the expression.

$25 x^{3} = 116985856 - \frac{2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.3.1.2

Cancel the common factor of $x$ .

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Step 10.2.3.1.2.1

Move the leading negative in $- \frac{2790528}{x}$ into the numerator.

$25 x^{3} = 116985856 + \frac{- 2790528}{x} x^{2} + 16641 x^{2}$

Step 10.2.3.1.2.2

Factor $x$ out of $x^{2}$ .

$25 x^{3} = 116985856 + \frac{- 2790528}{x} (x \cdot x) + 16641 x^{2}$

Step 10.2.3.1.2.3

Cancel the common factor.

$25 x^{3} = 116985856 + \frac{- 2790528}{x} (x \cdot x) + 16641 x^{2}$

Step 10.2.3.1.2.4

Rewrite the expression.

$25 x^{3} = 116985856 - 2790528 x + 16641 x^{2}$

Step 10.3

Solve the equation.

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Step 10.3.1

Move all the expressions to the left side of the equation.

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Step 10.3.1.1

Subtract $116985856$ from both sides of the equation.

$25 x^{3} - 116985856 = - 2790528 x + 16641 x^{2}$

Step 10.3.1.2

Add $2790528 x$ to both sides of the equation.

$25 x^{3} - 116985856 + 2790528 x = 16641 x^{2}$

Step 10.3.1.3

Subtract $16641 x^{2}$ from both sides of the equation.

$25 x^{3} - 116985856 + 2790528 x - 16641 x^{2} = 0$

Step 10.3.2

Factor the left side of the equation.

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Step 10.3.2.1

Reorder terms.

$25 x^{3} - 16641 x^{2} + 2790528 x - 116985856 = 0$

Step 10.3.2.2

Factor $25 x^{3} - 16641 x^{2} + 2790528 x - 116985856$ using the rational roots test.

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Step 10.3.2.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 116985856, \pm 2, \pm 58492928, \pm 4, \pm 29246464, \pm 8, \pm 14623232, \pm 13, \pm 8998912, \pm 16, \pm 7311616, \pm 26, \pm 4499456, \pm 32, \pm 3655808, \pm 52, \pm 2249728, \pm 64, \pm 1827904, \pm 104, \pm 1124864, \pm 128, \pm 913952, \pm 169, \pm 692224, \pm 208, \pm 562432, \pm 256, \pm 456976, \pm 338, \pm 346112, \pm 416, \pm 281216, \pm 512, \pm 228488, \pm 676, \pm 173056, \pm 832, \pm 140608, \pm 1024, \pm 114244, \pm 1352, \pm 86528, \pm 1664, \pm 70304, \pm 2048, \pm 57122, \pm 2197, \pm 53248, \pm 2704, \pm 43264, \pm 3328, \pm 35152, \pm 4096, \pm 28561, \pm 4394, \pm 26624, \pm 5408, \pm 21632, \pm 6656, \pm 17576, \pm 8788, \pm 13312, \pm 10816$

$q = \pm 1, \pm 25, \pm 5$

Step 10.3.2.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.04, \pm 0.2, \pm 116985856, \pm 4679434.24, \pm 23397171.2, \pm 2, \pm 0.08, \pm 0.4, \pm 58492928, \pm 2339717.12, \pm 11698585.6, \pm 4, \pm 0.16, \pm 0.8, \pm 29246464, \pm 1169858.56, \pm 5849292.8, \pm 8, \pm 0.32, \pm 1.6, \pm 14623232, \pm 584929.28, \pm 2924646.4, \pm 13, \pm 0.52, \pm 2.6, \pm 8998912, \pm 359956.48, \pm 1799782.4, \pm 16, \pm 0.64, \pm 3.2, \pm 7311616, \pm 292464.64, \pm 1462323.2, \pm 26, \pm 1.04, \pm 5.2, \pm 4499456, \pm 179978.24, \pm 899891.2, \pm 32, \pm 1.28, \pm 6.4, \pm 3655808, \pm 146232.32, \pm 731161.6, \pm 52, \pm 2.08, \pm 10.4, \pm 2249728, \pm 89989.12, \pm 449945.6, \pm 64, \pm 2.56, \pm 12.8, \pm 1827904, \pm 73116.16, \pm 365580.8, \pm 104, \pm 4.16, \pm 20.8, \pm 1124864, \pm 44994.56, \pm 224972.8, \pm 128, \pm 5.12, \pm 25.6, \pm 913952, \pm 36558.08, \pm 182790.4, \pm 169, \pm 6.76, \pm 33.8, \pm 692224, \pm 27688.96, \pm 138444.8, \pm 208, \pm 8.32, \pm 41.6, \pm 562432, \pm 22497.28, \pm 112486.4, \pm 256, \pm 10.24, \pm 51.2, \pm 456976, \pm 18279.04, \pm 91395.2, \pm 338, \pm 13.52, \pm 67.6, \pm 346112, \pm 13844.48, \pm 69222.4, \pm 416, \pm 16.64, \pm 83.2, \pm 281216, \pm 11248.64, \pm 56243.2, \pm 512, \pm 20.48, \pm 102.4, \pm 228488, \pm 9139.52, \pm 45697.6, \pm 676, \pm 27.04, \pm 135.2, \pm 173056, \pm 6922.24, \pm 34611.2, \pm 832, \pm 33.28, \pm 166.4, \pm 140608, \pm 5624.32, \pm 28121.6, \pm 1024, \pm 40.96, \pm 204.8, \pm 114244, \pm 4569.76, \pm 22848.8, \pm 1352, \pm 54.08, \pm 270.4, \pm 86528, \pm 3461.12, \pm 17305.6, \pm 1664, \pm 66.56, \pm 332.8, \pm 70304, \pm 2812.16, \pm 14060.8, \pm 2048, \pm 81.92, \pm 409.6, \pm 57122, \pm 2284.88, \pm 11424.4, \pm 2197, \pm 87.88, \pm 439.4, \pm 53248, \pm 2129.92, \pm 10649.6, \pm 2704, \pm 108.16, \pm 540.8, \pm 43264, \pm 1730.56, \pm 8652.8, \pm 3328, \pm 133.12, \pm 665.6, \pm 35152, \pm 1406.08, \pm 7030.4, \pm 4096, \pm 163.84, \pm 819.2, \pm 28561, \pm 1142.44, \pm 5712.2, \pm 4394, \pm 175.76, \pm 878.8, \pm 26624, \pm 1064.96, \pm 5324.8, \pm 5408, \pm 216.32, \pm 1081.6, \pm 21632, \pm 865.28, \pm 4326.4, \pm 6656, \pm 266.24, \pm 1331.2, \pm 17576, \pm 703.04, \pm 3515.2, \pm 8788, \pm 351.52, \pm 1757.6, \pm 13312, \pm 532.48, \pm 2662.4, \pm 10816, \pm 432.64, \pm 2163.2$

Step 10.3.2.2.3

Substitute $64$ and simplify the expression. In this case, the expression is equal to $0$ so $64$ is a root of the polynomial.

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Step 10.3.2.2.3.1

Substitute $64$ into the polynomial.

$25 \cdot 64^{3} - 16641 \cdot 64^{2} + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.2

Raise $64$ to the power of $3$ .

$25 \cdot 262144 - 16641 \cdot 64^{2} + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.3

Multiply $25$ by $262144$ .

$6553600 - 16641 \cdot 64^{2} + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.4

Raise $64$ to the power of $2$ .

$6553600 - 16641 \cdot 4096 + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.5

Multiply $- 16641$ by $4096$ .

$6553600 - 68161536 + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.6

Subtract $68161536$ from $6553600$ .

$- 61607936 + 2790528 \cdot 64 - 116985856$

Step 10.3.2.2.3.7

Multiply $2790528$ by $64$ .

$- 61607936 + 178593792 - 116985856$

Step 10.3.2.2.3.8

Add $- 61607936$ and $178593792$ .

$116985856 - 116985856$

Step 10.3.2.2.3.9

Subtract $116985856$ from $116985856$ .

$0$

Step 10.3.2.2.4

Since $64$ is a known root, divide the polynomial by $x - 64$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{25 x^{3} - 16641 x^{2} + 2790528 x - 116985856}{x - 64}$

Step 10.3.2.2.5

Divide $25 x^{3} - 16641 x^{2} + 2790528 x - 116985856$ by $x - 64$ .

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Step 10.3.2.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$64$

$25 x^{3}$

$16641 x^{2}$

$2790528 x$

$116985856$

Step 10.3.2.2.5.2

Divide the highest order term in the dividend $25 x^{3}$ by the highest order term in divisor $x$ .

					$25 x^{2}$
	$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$

Step 10.3.2.2.5.3

Multiply the new quotient term by the divisor.

				$25 x^{2}$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			+	$25 x^{3}$	-	$1600 x^{2}$

Step 10.3.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{3} - 1600 x^{2}$

				$25 x^{2}$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$

Step 10.3.2.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$25 x^{2}$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$

Step 10.3.2.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$25 x^{2}$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$

Step 10.3.2.2.5.7

Divide the highest order term in the dividend $- 15041 x^{2}$ by the highest order term in divisor $x$ .

				$25 x^{2}$	-	$15041 x$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$

Step 10.3.2.2.5.8

Multiply the new quotient term by the divisor.

				$25 x^{2}$	-	$15041 x$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					-	$15041 x^{2}$	+	$962624 x$

Step 10.3.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15041 x^{2} + 962624 x$

				$25 x^{2}$	-	$15041 x$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$

Step 10.3.2.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$25 x^{2}$	-	$15041 x$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$

Step 10.3.2.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$25 x^{2}$	-	$15041 x$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$	-	$116985856$

Step 10.3.2.2.5.12

Divide the highest order term in the dividend $1827904 x$ by the highest order term in divisor $x$ .

				$25 x^{2}$	-	$15041 x$	+	$1827904$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$	-	$116985856$

Step 10.3.2.2.5.13

Multiply the new quotient term by the divisor.

				$25 x^{2}$	-	$15041 x$	+	$1827904$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$	-	$116985856$
							+	$1827904 x$	-	$116985856$

Step 10.3.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $1827904 x - 116985856$

				$25 x^{2}$	-	$15041 x$	+	$1827904$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$	-	$116985856$
							-	$1827904 x$	+	$116985856$

Step 10.3.2.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$25 x^{2}$	-	$15041 x$	+	$1827904$
$x$	-	$64$		$25 x^{3}$	-	$16641 x^{2}$	+	$2790528 x$	-	$116985856$
			-	$25 x^{3}$	+	$1600 x^{2}$
					-	$15041 x^{2}$	+	$2790528 x$
					+	$15041 x^{2}$	-	$962624 x$
							+	$1827904 x$	-	$116985856$
							-	$1827904 x$	+	$116985856$
										$0$

Step 10.3.2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$25 x^{2} - 15041 x + 1827904$

Step 10.3.2.2.6

Write $25 x^{3} - 16641 x^{2} + 2790528 x - 116985856$ as a set of factors.

$(x - 64) (25 x^{2} - 15041 x + 1827904) = 0$

Step 10.3.2.3

Factor.

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Step 10.3.2.3.1

Factor by grouping.

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Step 10.3.2.3.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 25 \cdot 1827904 = 45697600$ and whose sum is $b = - 15041$ .

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Step 10.3.2.3.1.1.1

Factor $- 15041$ out of $- 15041 x$ .

$(x - 64) (25 x^{2} - 15041 x + 1827904) = 0$

Step 10.3.2.3.1.1.2

Rewrite $- 15041$ as $- 4225$ plus $- 10816$

$(x - 64) (25 x^{2} + (- 4225 - 10816) x + 1827904) = 0$

Step 10.3.2.3.1.1.3

Apply the distributive property.

$(x - 64) (25 x^{2} - 4225 x - 10816 x + 1827904) = 0$

Step 10.3.2.3.1.2

Factor out the greatest common factor from each group.

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Step 10.3.2.3.1.2.1

Group the first two terms and the last two terms.

$(x - 64) ((25 x^{2} - 4225 x) - 10816 x + 1827904) = 0$

Step 10.3.2.3.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x - 64) (25 x (x - 169) - 10816 (x - 169)) = 0$

Step 10.3.2.3.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 169$ .

$(x - 64) ((x - 169) (25 x - 10816)) = 0$

Step 10.3.2.3.2

Remove unnecessary parentheses.

$(x - 64) (x - 169) (25 x - 10816) = 0$

Step 10.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 64 = 0$

$x - 169 = 0$

$25 x - 10816 = 0$

Step 10.3.4

Set $x - 64$ equal to $0$ and solve for $x$ .

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Step 10.3.4.1

Set $x - 64$ equal to $0$ .

$x - 64 = 0$

Step 10.3.4.2

Add $64$ to both sides of the equation.

$x = 64$

Step 10.3.5

Set $x - 169$ equal to $0$ and solve for $x$ .

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Step 10.3.5.1

Set $x - 169$ equal to $0$ .

$x - 169 = 0$

Step 10.3.5.2

Add $169$ to both sides of the equation.

$x = 169$

Step 10.3.6

Set $25 x - 10816$ equal to $0$ and solve for $x$ .

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Step 10.3.6.1

Set $25 x - 10816$ equal to $0$ .

$25 x - 10816 = 0$

Step 10.3.6.2

Solve $25 x - 10816 = 0$ for $x$ .

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Step 10.3.6.2.1

Add $10816$ to both sides of the equation.

$25 x = 10816$

Step 10.3.6.2.2

Divide each term in $25 x = 10816$ by $25$ and simplify.

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Step 10.3.6.2.2.1

Divide each term in $25 x = 10816$ by $25$ .

$\frac{25 x}{25} = \frac{10816}{25}$

Step 10.3.6.2.2.2

Simplify the left side.

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Step 10.3.6.2.2.2.1

Cancel the common factor of $25$ .

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Step 10.3.6.2.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{10816}{25}$

Step 10.3.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10816}{25}$

Step 10.3.7

The final solution is all the values that make $(x - 64) (x - 169) (25 x - 10816) = 0$ true.

$x = 64, 169, \frac{10816}{25}$

Step 11

Exclude the solutions that do not make $\log_{8} (5 x) - \log_{8} (13 + \sqrt{x}) = 2$ true.

$x = \frac{10816}{25}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10816}{25}$

Decimal Form:

$x = 432.64$

Mixed Number Form:

$x = 432 \frac{16}{25}$