Basic Math Examples

$\frac{\sqrt{\sqrt[3]{7}} \cdot \sqrt{7}}{\sqrt[3]{7^{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

Combine using the product rule for radicals.

$\frac{\sqrt{\sqrt[3]{7} \cdot 7}}{\sqrt[3]{7^{2}}}$

Step 1.2

Move $7$ to the left of $\sqrt[3]{7}$ .

$\frac{\sqrt{7 \sqrt[3]{7}}}{\sqrt[3]{7^{2}}}$

Step 2

Raise $7$ to the power of $2$ .

$\frac{\sqrt{7 \sqrt[3]{7}}}{\sqrt[3]{49}}$

Step 3

Multiply $\frac{\sqrt{7 \sqrt[3]{7}}}{\sqrt[3]{49}}$ by $\frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$ .

$\frac{\sqrt{7 \sqrt[3]{7}}}{\sqrt[3]{49}} \cdot \frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{7 \sqrt[3]{7}}}{\sqrt[3]{49}}$ by $\frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$ .

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{\sqrt[3]{49} {\sqrt[3]{49}}^{2}}$

Step 4.2

Raise $\sqrt[3]{49}$ to the power of $1$ .

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{1} {\sqrt[3]{49}}^{2}}$

Step 4.3

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

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{1 + 2}}$

Step 4.4

Add $1$ and $2$ .

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{3}}$

Step 4.5

Rewrite ${\sqrt[3]{49}}^{3}$ as $49$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{49}$ as $49^{\frac{1}{3}}$ .

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{{(49^{\frac{1}{3}})}^{3}}$

Step 4.5.2

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

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{49^{\frac{1}{3} \cdot 3}}$

Step 4.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{49^{\frac{3}{3}}}$

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{49^{\frac{3}{3}}}$

Step 4.5.4.2

Rewrite the expression.

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{49^{1}}$

Step 4.5.5

Evaluate the exponent.

$\frac{\sqrt{7 \sqrt[3]{7}} {\sqrt[3]{49}}^{2}}{49}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{49}}^{2}$ as $\sqrt[3]{49^{2}}$ .

$\frac{\sqrt{7 \sqrt[3]{7}} \sqrt[3]{49^{2}}}{49}$

Step 5.2

Raise $49$ to the power of $2$ .

$\frac{\sqrt{7 \sqrt[3]{7}} \sqrt[3]{2401}}{49}$

Step 5.3

Rewrite $2401$ as $7^{3} \cdot 7$ .

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Step 5.3.1

Factor $343$ out of $2401$ .

$\frac{\sqrt{7 \sqrt[3]{7}} \sqrt[3]{343 (7)}}{49}$

Step 5.3.2

Rewrite $343$ as $7^{3}$ .

$\frac{\sqrt{7 \sqrt[3]{7}} \sqrt[3]{7^{3} \cdot 7}}{49}$

Step 5.4

Pull terms out from under the radical.

$\frac{\sqrt{7 \sqrt[3]{7}} \cdot 7 \sqrt[3]{7}}{49}$

Step 5.5

Combine exponents.

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Step 5.5.1

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

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Step 5.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 \sqrt[3]{7}}$ as ${(7 \sqrt[3]{7})}^{\frac{1}{2}}$ .

$\frac{7 ({(7 \sqrt[3]{7})}^{\frac{1}{2}} \sqrt[3]{7})}{49}$

Step 5.5.1.2

Rewrite ${(7 \sqrt[3]{7})}^{\frac{1}{2}}$ as ${(7 \sqrt[3]{7})}^{\frac{3}{6}}$ .

$\frac{7 ({(7 \sqrt[3]{7})}^{\frac{3}{6}} \sqrt[3]{7})}{49}$

Step 5.5.1.3

Rewrite ${(7 \sqrt[3]{7})}^{\frac{3}{6}}$ as $\sqrt[6]{{(7 \sqrt[3]{7})}^{3}}$ .

$\frac{7 (\sqrt[6]{{(7 \sqrt[3]{7})}^{3}} \sqrt[3]{7})}{49}$

Step 5.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{7 (\sqrt[6]{{(7 \sqrt[3]{7})}^{3}} \cdot 7^{\frac{1}{3}})}{49}$

Step 5.5.1.5

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{2}{6}}$ .

$\frac{7 (\sqrt[6]{{(7 \sqrt[3]{7})}^{3}} \cdot 7^{\frac{2}{6}})}{49}$

Step 5.5.1.6

Rewrite $7^{\frac{2}{6}}$ as $\sqrt[6]{7^{2}}$ .

$\frac{7 (\sqrt[6]{{(7 \sqrt[3]{7})}^{3}} \sqrt[6]{7^{2}})}{49}$

Step 5.5.2

Combine using the product rule for radicals.

$\frac{7 \sqrt[6]{{(7 \sqrt[3]{7})}^{3} \cdot 7^{2}}}{49}$

Step 5.5.3

Apply the product rule to $7 \sqrt[3]{7}$ .

$\frac{7 \sqrt[6]{7^{3} {\sqrt[3]{7}}^{3} \cdot 7^{2}}}{49}$

Step 5.5.4

Multiply $7^{3}$ by $7^{2}$ by adding the exponents.

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Step 5.5.4.1

Move $7^{2}$ .

$\frac{7 \sqrt[6]{7^{2} \cdot 7^{3} {\sqrt[3]{7}}^{3}}}{49}$

Step 5.5.4.2

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

$\frac{7 \sqrt[6]{7^{2 + 3} {\sqrt[3]{7}}^{3}}}{49}$

Step 5.5.4.3

Add $2$ and $3$ .

$\frac{7 \sqrt[6]{7^{5} {\sqrt[3]{7}}^{3}}}{49}$

Step 5.5.5

Raise $7$ to the power of $5$ .

$\frac{7 \sqrt[6]{16807 {\sqrt[3]{7}}^{3}}}{49}$

Step 5.5.6

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 5.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{7 \sqrt[6]{16807 {(7^{\frac{1}{3}})}^{3}}}{49}$

Step 5.5.6.2

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

$\frac{7 \sqrt[6]{16807 \cdot 7^{\frac{1}{3} \cdot 3}}}{49}$

Step 5.5.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{7 \sqrt[6]{16807 \cdot 7^{\frac{3}{3}}}}{49}$

Step 5.5.6.4

Cancel the common factor of $3$ .

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Step 5.5.6.4.1

Cancel the common factor.

$\frac{7 \sqrt[6]{16807 \cdot 7^{\frac{3}{3}}}}{49}$

Step 5.5.6.4.2

Rewrite the expression.

$\frac{7 \sqrt[6]{16807 \cdot 7^{1}}}{49}$

Step 5.5.6.5

Evaluate the exponent.

$\frac{7 \sqrt[6]{16807 \cdot 7}}{49}$

Step 5.5.7

Multiply $16807$ by $7$ .

$\frac{7 \sqrt[6]{117649}}{49}$

Step 5.6

Rewrite $117649$ as $7^{6}$ .

$\frac{7 \sqrt[6]{7^{6}}}{49}$

Step 5.7

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

$\frac{7 \cdot 7}{49}$

Step 6

Simplify the expression.

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Step 6.1

Multiply $7$ by $7$ .

$\frac{49}{49}$

Step 6.2

Divide $49$ by $49$ .

$1$