Simplify ( cube root of x^7)/( square root of x^3)

$\frac{\sqrt[3]{x^{7}}}{\sqrt{x^{3}}}$

Step 1

Simplify the numerator.

Rewrite $x^{7}$ as ${(x^{2})}^{3} x$ .

Factor out $x^{6}$ .

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

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[3]{x}}{\sqrt{x^{3}}}$

Step 2

Simplify the denominator.

Factor out $x^{2}$ .

$\frac{x^{2} \sqrt[3]{x}}{\sqrt{x^{2} x}}$

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[3]{x}}{x \sqrt{x}}$

Step 3

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

Factor $x$ out of $x^{2} \sqrt[3]{x}$ .

$\frac{x (x \sqrt[3]{x})}{x \sqrt{x}}$

Cancel the common factors.

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

$\frac{x (x \sqrt[3]{x})}{x (\sqrt{x})}$

Cancel the common factor.

$\frac{x (x \sqrt[3]{x})}{x \sqrt{x}}$

Rewrite the expression.

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

Step 4

Multiply $\frac{x \sqrt[3]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{x \sqrt[3]{x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{x \sqrt[3]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

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

$\frac{x \sqrt[3]{x} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

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

$\frac{x \sqrt[3]{x} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

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

$\frac{x \sqrt[3]{x} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{x \sqrt[3]{x} \sqrt{x}}{{\sqrt{x}}^{2}}$

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

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

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

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

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

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

$\frac{x \sqrt[3]{x} \sqrt{x}}{x^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x \sqrt[3]{x} \sqrt{x}}{x^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{x \sqrt[3]{x} \sqrt{x}}{x^{1}}$

Simplify.

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

Step 6

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $\sqrt[3]{x} \sqrt{x}$ by $1$ .

$\sqrt[3]{x} \sqrt{x}$

Step 7

Multiply $\sqrt[3]{x} \sqrt{x}$ .

Rewrite the expression using the least common index of $6$ .

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

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

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$x^{\frac{2}{6}} \sqrt{x}$

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$\sqrt[6]{x^{2}} \sqrt{x}$

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

$\sqrt[6]{x^{2}} x^{\frac{1}{2}}$

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$\sqrt[6]{x^{2}} x^{\frac{3}{6}}$

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$\sqrt[6]{x^{2}} \sqrt[6]{x^{3}}$

Combine using the product rule for radicals.

$\sqrt[6]{x^{2} x^{3}}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

$\sqrt[6]{x^{2 + 3}}$

Add $2$ and $3$ .

$\sqrt[6]{x^{5}}$

Simplify ( cube root of x^7)/( square root of x^3)

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