Algebra Examples

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

Step 1

Multiply $\sqrt[7]{7} \sqrt[3]{7}$ .

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Step 1.1

Rewrite the expression using the least common index of $21$ .

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Step 1.1.1

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

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

Step 1.1.2

Rewrite $7^{\frac{1}{7}}$ as $7^{\frac{3}{21}}$ .

$7^{\frac{3}{21}} \sqrt[3]{7}$

Step 1.1.3

Rewrite $7^{\frac{3}{21}}$ as $\sqrt[21]{7^{3}}$ .

$\sqrt[21]{7^{3}} \sqrt[3]{7}$

Step 1.1.4

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

$\sqrt[21]{7^{3}} \cdot 7^{\frac{1}{3}}$

Step 1.1.5

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{7}{21}}$ .

$\sqrt[21]{7^{3}} \cdot 7^{\frac{7}{21}}$

Step 1.1.6

Rewrite $7^{\frac{7}{21}}$ as $\sqrt[21]{7^{7}}$ .

$\sqrt[21]{7^{3}} \sqrt[21]{7^{7}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[21]{7^{3} \cdot 7^{7}}$

Step 1.3

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

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Step 1.3.1

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

$\sqrt[21]{7^{3 + 7}}$

Step 1.3.2

Add $3$ and $7$ .

$\sqrt[21]{7^{10}}$

Step 2

Raise $7$ to the power of $10$ .

$\sqrt[21]{282475249}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[21]{282475249}$

Decimal Form:

$2.52596679 \dots$