Algebra Examples

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

Step 1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 2

Any root of $1$ is $1$ .

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

Step 3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4

Combine fractions.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}})}^{3} {(\frac{2}{\sqrt{3}})}^{3}$

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Add $1$ and $1$ .

${(\frac{\sqrt{2}}{{\sqrt{2}}^{2}})}^{3} {(\frac{2}{\sqrt{3}})}^{3}$

Step 4.1.6

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

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

${(\frac{\sqrt{2}}{2^{\frac{2}{2}}})}^{3} {(\frac{2}{\sqrt{3}})}^{3}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

${(\frac{\sqrt{2}}{2^{\frac{2}{2}}})}^{3} {(\frac{2}{\sqrt{3}})}^{3}$

Step 4.1.6.4.2

Rewrite the expression.

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

Step 4.1.6.5

Evaluate the exponent.

${(\frac{\sqrt{2}}{2})}^{3} {(\frac{2}{\sqrt{3}})}^{3}$

Step 4.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{{\sqrt{2}}^{3}}{2^{3}} {(\frac{2}{\sqrt{3}})}^{3}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\frac{\sqrt{2^{3}}}{2^{3}} {(\frac{2}{\sqrt{3}})}^{3}$

Step 5.2

Raise $2$ to the power of $3$ .

$\frac{\sqrt{8}}{2^{3}} {(\frac{2}{\sqrt{3}})}^{3}$

Step 5.3

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

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Step 5.3.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{2^{3}} {(\frac{2}{\sqrt{3}})}^{3}$

Step 5.3.2

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

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

Step 5.4

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{2^{3}} {(\frac{2}{\sqrt{3}})}^{3}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Raise $2$ to the power of $3$ .

$\frac{2 \sqrt{2}}{8} {(\frac{2}{\sqrt{3}})}^{3}$

Step 6.2

Cancel the common factor of $2$ and $8$ .

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Step 6.2.1

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

$\frac{2 (\sqrt{2})}{8} {(\frac{2}{\sqrt{3}})}^{3}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \sqrt{2}}{2 \cdot 4} {(\frac{2}{\sqrt{3}})}^{3}$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \sqrt{2}}{2 \cdot 4} {(\frac{2}{\sqrt{3}})}^{3}$

Step 6.2.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{4} {(\frac{2}{\sqrt{3}})}^{3}$

Step 7

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

$\frac{\sqrt{2}}{4} {(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{3}$

Step 8

Combine and simplify the denominator.

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Step 8.1

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

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{3}$

Step 8.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{3}$

Step 8.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{3}$

Step 8.4

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

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{3}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{3}$

Step 8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{3}$

Step 8.6.2

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

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{3}$

Step 8.6.3

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

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{3}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{3}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{3^{1}})}^{3}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{4} {(\frac{2 \sqrt{3}}{3})}^{3}$

Step 9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$\frac{\sqrt{2}}{4} \cdot \frac{{(2 \sqrt{3})}^{3}}{3^{3}}$

Step 9.2

Apply the product rule to $2 \sqrt{3}$ .

$\frac{\sqrt{2}}{4} \cdot \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}}$

Step 10

Combine.

$\frac{\sqrt{2} (2^{3} {\sqrt{3}}^{3})}{4 \cdot 3^{3}}$

Step 11

Simplify the numerator.

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Step 11.1

Raise $2$ to the power of $3$ .

$\frac{\sqrt{2} \cdot 8 {\sqrt{3}}^{3}}{4 \cdot 3^{3}}$

Step 11.2

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{\sqrt{2} \cdot 8 \sqrt{3^{3}}}{4 \cdot 3^{3}}$

Step 11.3

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

$\frac{\sqrt{2} \cdot 8 \sqrt{27}}{4 \cdot 3^{3}}$

Step 11.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 11.4.1

Factor $9$ out of $27$ .

$\frac{\sqrt{2} \cdot 8 \sqrt{9 (3)}}{4 \cdot 3^{3}}$

Step 11.4.2

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

$\frac{\sqrt{2} \cdot 8 \sqrt{3^{2} \cdot 3}}{4 \cdot 3^{3}}$

Step 11.5

Pull terms out from under the radical.

$\frac{\sqrt{2} \cdot 8 \cdot 3 \sqrt{3}}{4 \cdot 3^{3}}$

Step 11.6

Combine exponents.

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Step 11.6.1

Multiply $3$ by $8$ .

$\frac{\sqrt{2} \cdot 24 \sqrt{3}}{4 \cdot 3^{3}}$

Step 11.6.2

Combine using the product rule for radicals.

$\frac{24 \sqrt{2 \cdot 3}}{4 \cdot 3^{3}}$

Step 11.6.3

Multiply $2$ by $3$ .

$\frac{24 \sqrt{6}}{4 \cdot 3^{3}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

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

$\frac{24 \sqrt{6}}{4 \cdot 27}$

Step 12.2

Multiply $4$ by $27$ .

$\frac{24 \sqrt{6}}{108}$

Step 12.3

Cancel the common factor of $24$ and $108$ .

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Step 12.3.1

Factor $12$ out of $24 \sqrt{6}$ .

$\frac{12 (2 \sqrt{6})}{108}$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

Factor $12$ out of $108$ .

$\frac{12 (2 \sqrt{6})}{12 (9)}$

Step 12.3.2.2

Cancel the common factor.

$\frac{12 (2 \sqrt{6})}{12 \cdot 9}$

Step 12.3.2.3

Rewrite the expression.

$\frac{2 \sqrt{6}}{9}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{6}}{9}$

Decimal Form:

$0.54433105 \dots$