Algebra Examples

$\sqrt{7} \cdot \sqrt[9]{7}$

Step 1

Multiply $\sqrt{7} \sqrt[9]{7}$ .

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Step 1.1

Rewrite the expression using the least common index of $18$ .

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Step 1.1.1

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

$7^{\frac{1}{2}} \sqrt[9]{7}$

Step 1.1.2

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{9}{18}}$ .

$7^{\frac{9}{18}} \sqrt[9]{7}$

Step 1.1.3

Rewrite $7^{\frac{9}{18}}$ as $\sqrt[18]{7^{9}}$ .

$\sqrt[18]{7^{9}} \sqrt[9]{7}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{7}$ as $7^{\frac{1}{9}}$ .

$\sqrt[18]{7^{9}} \cdot 7^{\frac{1}{9}}$

Step 1.1.5

Rewrite $7^{\frac{1}{9}}$ as $7^{\frac{2}{18}}$ .

$\sqrt[18]{7^{9}} \cdot 7^{\frac{2}{18}}$

Step 1.1.6

Rewrite $7^{\frac{2}{18}}$ as $\sqrt[18]{7^{2}}$ .

$\sqrt[18]{7^{9}} \sqrt[18]{7^{2}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[18]{7^{9} \cdot 7^{2}}$

Step 1.3

Multiply $7^{9}$ by $7^{2}$ 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.

$\sqrt[18]{7^{9 + 2}}$

Step 1.3.2

Add $9$ and $2$ .

$\sqrt[18]{7^{11}}$

Step 2

Raise $7$ to the power of $11$ .

$\sqrt[18]{1977326743}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[18]{1977326743}$

Decimal Form:

$3.28434523 \dots$