Simplify cube root of 7^2* fifth root of 7^3

$\sqrt[3]{7^{2}} \cdot \sqrt[5]{7^{3}}$

Step 1

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

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

Step 2

Raise $7$ to the power of $3$ .

$\sqrt[3]{49} \cdot \sqrt[5]{343}$

Step 3

Multiply $\sqrt[3]{49} \sqrt[5]{343}$ .

Rewrite the expression using the least common index of $15$ .

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

$49^{\frac{1}{3}} \sqrt[5]{343}$

Rewrite $49^{\frac{1}{3}}$ as $49^{\frac{5}{15}}$ .

$49^{\frac{5}{15}} \sqrt[5]{343}$

Rewrite $49^{\frac{5}{15}}$ as $\sqrt[15]{49^{5}}$ .

$\sqrt[15]{49^{5}} \sqrt[5]{343}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{343}$ as $343^{\frac{1}{5}}$ .

$\sqrt[15]{49^{5}} \cdot 343^{\frac{1}{5}}$

Rewrite $343^{\frac{1}{5}}$ as $343^{\frac{3}{15}}$ .

$\sqrt[15]{49^{5}} \cdot 343^{\frac{3}{15}}$

Rewrite $343^{\frac{3}{15}}$ as $\sqrt[15]{343^{3}}$ .

$\sqrt[15]{49^{5}} \sqrt[15]{343^{3}}$

Combine using the product rule for radicals.

$\sqrt[15]{49^{5} \cdot 343^{3}}$

Rewrite $49$ as $7^{2}$ .

$\sqrt[15]{{(7^{2})}^{5} \cdot 343^{3}}$

Multiply the exponents in ${(7^{2})}^{5}$ .

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

$\sqrt[15]{7^{2 \cdot 5} \cdot 343^{3}}$

Multiply $2$ by $5$ .

$\sqrt[15]{7^{10} \cdot 343^{3}}$

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

$\sqrt[15]{7^{10} \cdot {(7^{3})}^{3}}$

Multiply the exponents in ${(7^{3})}^{3}$ .

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

$\sqrt[15]{7^{10} \cdot 7^{3 \cdot 3}}$

Multiply $3$ by $3$ .

$\sqrt[15]{7^{10} \cdot 7^{9}}$

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

$\sqrt[15]{7^{10 + 9}}$

Add $10$ and $9$ .

$\sqrt[15]{7^{19}}$

Step 4

Raise $7$ to the power of $19$ .

$\sqrt[15]{11398895185373100}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt[15]{11398895185373100}$

Decimal Form:

$11.76135929 \dots$

Simplify cube root of 7^2* fifth root of 7^3

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