Algebra Examples

$i \sqrt{\frac{8}{5}} \cdot i \sqrt{5} \cdot i \sqrt{\frac{9}{2}}$

Step 1

Rewrite $\sqrt{\frac{8}{5}}$ as $\frac{\sqrt{8}}{\sqrt{5}}$ .

$i \frac{\sqrt{8}}{\sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $8$ .

$i \frac{\sqrt{4 (2)}}{\sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 2.1.2

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

$i \frac{\sqrt{2^{2} \cdot 2}}{\sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 2.2

Pull terms out from under the radical.

$i \frac{2 \sqrt{2}}{\sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 3

Multiply $\frac{2 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$i (\frac{2 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$i \frac{2 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.2

Raise $\sqrt{5}$ to the power of $1$ .

$i \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.3

Raise $\sqrt{5}$ to the power of $1$ .

$i \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.4

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

$i \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.5

Add $1$ and $1$ .

$i \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$i \frac{2 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6.2

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

$i \frac{2 \sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6.3

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

$i \frac{2 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$i \frac{2 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6.4.2

Rewrite the expression.

$i \frac{2 \sqrt{2} \sqrt{5}}{5^{1}} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 4.6.5

Evaluate the exponent.

$i \frac{2 \sqrt{2} \sqrt{5}}{5} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$i \frac{2 \sqrt{5 \cdot 2}}{5} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 5.2

Multiply $5$ by $2$ .

$i \frac{2 \sqrt{10}}{5} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 6

Combine fractions.

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Step 6.1

Combine $i$ and $\frac{2 \sqrt{10}}{5}$ .

$\frac{i (2 \sqrt{10})}{5} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 6.2

Move $2$ to the left of $i$ .

$\frac{2 i \sqrt{10}}{5} \cdot (i \sqrt{5}) \cdot (i \sqrt{\frac{9}{2}})$

Step 7

Multiply $\frac{2 i \sqrt{10}}{5} (i \sqrt{5})$ .

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Step 7.1

Combine $i$ and $\frac{2 i \sqrt{10}}{5}$ .

$\frac{i (2 i \sqrt{10})}{5} \sqrt{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.2

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

$\frac{2 (i^{1} i) \sqrt{10}}{5} \sqrt{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.3

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

$\frac{2 (i^{1} i^{1}) \sqrt{10}}{5} \sqrt{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.4

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

$\frac{2 i^{1 + 1} \sqrt{10}}{5} \sqrt{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.5

Add $1$ and $1$ .

$\frac{2 i^{2} \sqrt{10}}{5} \sqrt{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.6

Combine $\frac{2 i^{2} \sqrt{10}}{5}$ and $\sqrt{5}$ .

$\frac{2 i^{2} \sqrt{10} \sqrt{5}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.7

Combine using the product rule for radicals.

$\frac{2 i^{2} \sqrt{5 \cdot 10}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 7.8

Multiply $5$ by $10$ .

$\frac{2 i^{2} \sqrt{50}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{2 \cdot - 1 \sqrt{50}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.2

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 8.2.1

Factor $25$ out of $50$ .

$\frac{2 \cdot - 1 \sqrt{25 (2)}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.2.2

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

$\frac{2 \cdot - 1 \sqrt{5^{2} \cdot 2}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.3

Pull terms out from under the radical.

$\frac{2 \cdot - 1 \cdot 5 \sqrt{2}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.4

Combine exponents.

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Step 8.4.1

Factor out negative.

$\frac{- (2 \cdot 5 \sqrt{2})}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.4.2

Multiply $2$ by $5$ .

$\frac{- (10 \sqrt{2})}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 8.4.3

Multiply $10$ by $- 1$ .

$\frac{- 10 \sqrt{2}}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 9

Cancel the common factor of $- 10$ and $5$ .

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Step 9.1

Factor $5$ out of $- 10 \sqrt{2}$ .

$\frac{5 (- 2 \sqrt{2})}{5} \cdot (i \sqrt{\frac{9}{2}})$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 2 \sqrt{2})}{5 (1)} \cdot (i \sqrt{\frac{9}{2}})$

Step 9.2.2

Cancel the common factor.

$\frac{5 (- 2 \sqrt{2})}{5 \cdot 1} \cdot (i \sqrt{\frac{9}{2}})$

Step 9.2.3

Rewrite the expression.

$\frac{- 2 \sqrt{2}}{1} \cdot (i \sqrt{\frac{9}{2}})$

Step 9.2.4

Divide $- 2 \sqrt{2}$ by $1$ .

$- 2 \sqrt{2} \cdot (i \sqrt{\frac{9}{2}})$

Step 10

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

$- 2 \sqrt{2} \cdot (i \frac{\sqrt{9}}{\sqrt{2}})$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite $9$ as $3^{2}$ .

$- 2 \sqrt{2} \cdot (i \frac{\sqrt{3^{2}}}{\sqrt{2}})$

Step 11.2

Pull terms out from under the radical, assuming positive real numbers.

$- 2 \sqrt{2} \cdot (i \frac{3}{\sqrt{2}})$

Step 12

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- 2 \sqrt{2} \cdot (i (\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 13.2

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})$

Step 13.3

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})$

Step 13.4

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Step 13.5

Add $1$ and $1$ .

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 13.6

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

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Step 13.6.1

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 13.6.2

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 13.6.3

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

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 13.6.4

Cancel the common factor of $2$ .

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Step 13.6.4.1

Cancel the common factor.

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 13.6.4.2

Rewrite the expression.

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{2^{1}})$

Step 13.6.5

Evaluate the exponent.

$- 2 \sqrt{2} \cdot (i \frac{3 \sqrt{2}}{2})$

Step 14

Combine $i$ and $\frac{3 \sqrt{2}}{2}$ .

$- 2 \sqrt{2} \cdot \frac{i (3 \sqrt{2})}{2}$

Step 15

Move $3$ to the left of $i$ .

$- 2 \sqrt{2} \cdot \frac{3 i \sqrt{2}}{2}$

Step 16

Cancel the common factor of $2$ .

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Step 16.1

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

$2 (- \sqrt{2}) \cdot \frac{3 i \sqrt{2}}{2}$

Step 16.2

Cancel the common factor.

$2 (- \sqrt{2}) \cdot \frac{3 i \sqrt{2}}{2}$

Step 16.3

Rewrite the expression.

$- \sqrt{2} \cdot (3 i \sqrt{2})$

Step 17

Multiply $3$ by $- 1$ .

$- 3 \sqrt{2} \cdot (i \sqrt{2})$

Step 18

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

$- 3 ({\sqrt{2}}^{1} \sqrt{2}) \cdot i$

Step 19

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

$- 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) \cdot i$

Step 20

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

$- 3 {\sqrt{2}}^{1 + 1} \cdot i$

Step 21

Add $1$ and $1$ .

$- 3 {\sqrt{2}}^{2} \cdot i$

Step 22

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

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Step 22.1

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

$- 3 {(2^{\frac{1}{2}})}^{2} \cdot i$

Step 22.2

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

$- 3 \cdot 2^{\frac{1}{2} \cdot 2} \cdot i$

Step 22.3

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

$- 3 \cdot 2^{\frac{2}{2}} \cdot i$

Step 22.4

Cancel the common factor of $2$ .

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Step 22.4.1

Cancel the common factor.

$- 3 \cdot 2^{\frac{2}{2}} \cdot i$

Step 22.4.2

Rewrite the expression.

$- 3 \cdot 2^{1} \cdot i$

Step 22.5

Evaluate the exponent.

$- 3 \cdot 2 \cdot i$

Step 23

Multiply $- 3$ by $2$ .

$- 6 i$