Finite Math Examples

$\sqrt{\frac{x^{4}}{3} \cdot \sqrt{\frac{x^{5}}{3}}}$

Step 1

Combine $\frac{x^{4}}{3}$ and $\sqrt{\frac{x^{5}}{3}}$ .

$\sqrt{\frac{x^{4} \sqrt{\frac{x^{5}}{3}}}{3}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{x^{5}}{3}}$ as $\frac{\sqrt{x^{5}}}{\sqrt{3}}$ .

$\sqrt{\frac{x^{4} \frac{\sqrt{x^{5}}}{\sqrt{3}}}{3}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 2.2.1.1

Factor out $x^{4}$ .

$\sqrt{\frac{x^{4} \frac{\sqrt{x^{4} x}}{\sqrt{3}}}{3}}$

Step 2.2.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.2.2

Pull terms out from under the radical.

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

Step 2.3

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

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

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

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

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

Step 2.4.2

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

$\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{3}}$

Step 2.4.3

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

$\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{3}}$

Step 2.4.4

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

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

Step 2.4.5

Add $1$ and $1$ .

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

Step 2.4.6

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

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Step 2.4.6.1

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

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

Step 2.4.6.2

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

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

Step 2.4.6.3

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

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

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

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

Step 2.4.6.4.2

Rewrite the expression.

$\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x} \sqrt{3}}{3^{1}}}{3}}$

Step 2.4.6.5

Evaluate the exponent.

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

Step 2.5

Combine using the product rule for radicals.

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

Step 3

Combine $x^{4}$ and $\frac{x^{2} \sqrt{3 x}}{3}$ .

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

Step 4

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 4.1

Move $x^{2}$ .

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

Step 4.2

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

$\sqrt{\frac{\frac{x^{2 + 4} \sqrt{3 x}}{3}}{3}}$

Step 4.3

Add $2$ and $4$ .

$\sqrt{\frac{\frac{x^{6} \sqrt{3 x}}{3}}{3}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{x^{6} \sqrt{3 x}}{3} \cdot \frac{1}{3}}$

Step 6

Combine.

$\sqrt{\frac{x^{6} \sqrt{3 x} \cdot 1}{3 \cdot 3}}$

Step 7

Multiply.

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Step 7.1

Multiply $x^{6}$ by $1$ .

$\sqrt{\frac{x^{6} \sqrt{3 x}}{3 \cdot 3}}$

Step 7.2

Multiply $3$ by $3$ .

$\sqrt{\frac{x^{6} \sqrt{3 x}}{9}}$

Step 8

Rewrite $\frac{x^{6} \sqrt{3 x}}{9}$ as ${(\frac{x^{3}}{3})}^{2} \sqrt{3 x}$ .

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Step 8.1

Factor the perfect power ${(x^{3})}^{2}$ out of $x^{6} \sqrt{3 x}$ .

$\sqrt{\frac{{(x^{3})}^{2} \sqrt{3 x}}{9}}$

Step 8.2

Factor the perfect power $3^{2}$ out of $9$ .

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

Step 8.3

Rearrange the fraction $\frac{{(x^{3})}^{2} \sqrt{3 x}}{3^{2} \cdot 1}$ .

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

Step 9

Pull terms out from under the radical.

$\frac{x^{3}}{3} \sqrt{\sqrt{3 x}}$

Step 10

Rewrite $\sqrt{\sqrt{3 x}}$ as $\sqrt[4]{3 x}$ .

$\frac{x^{3}}{3} \sqrt[4]{3 x}$

Step 11

Combine $\frac{x^{3}}{3}$ and $\sqrt[4]{3 x}$ .

$\frac{x^{3} \sqrt[4]{3 x}}{3}$