Algebra Examples

$\sqrt[5]{7^{3}} \cdot \sqrt[3]{7^{4}}$

Step 1

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

$\sqrt[5]{343} \cdot \sqrt[3]{7^{4}}$

Step 2

Raise $7$ to the power of $4$ .

$\sqrt[5]{343} \cdot \sqrt[3]{2401}$

Step 3

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

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Step 3.1

Factor $343$ out of $2401$ .

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

Step 3.2

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $\sqrt[5]{343} (7 \sqrt[3]{7})$ .

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Step 5.1

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

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Step 5.1.1

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

$7 (343^{\frac{1}{5}} \sqrt[3]{7})$

Step 5.1.2

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

$7 (343^{\frac{3}{15}} \sqrt[3]{7})$

Step 5.1.3

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

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

Step 5.1.4

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

$7 (\sqrt[15]{343^{3}} \cdot 7^{\frac{1}{3}})$

Step 5.1.5

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{5}{15}}$ .

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

Step 5.1.6

Rewrite $7^{\frac{5}{15}}$ as $\sqrt[15]{7^{5}}$ .

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Multiply $3$ by $3$ .

$7 \sqrt[15]{7^{9} \cdot 7^{5}}$

Step 5.5

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

$7 \sqrt[15]{7^{9 + 5}}$

Step 5.6

Add $9$ and $5$ .

$7 \sqrt[15]{7^{14}}$

Step 6

Raise $7$ to the power of $14$ .

$7 \sqrt[15]{678223072849}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$7 \sqrt[15]{678223072849}$

Decimal Form:

$43.03840922 \dots$