Simplify ( square root of 10)/( fourth root of 8)

$\frac{\sqrt{10}}{\sqrt[4]{8}}$

Step 1

Multiply $\frac{\sqrt{10}}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{\sqrt{10}}{\sqrt[4]{8}} \cdot \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt{10}}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{\sqrt[4]{8} {\sqrt[4]{8}}^{3}}$

Raise $\sqrt[4]{8}$ to the power of $1$ .

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1} {\sqrt[4]{8}}^{3}}$

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

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{4}}$

Rewrite ${\sqrt[4]{8}}^{4}$ as $8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{{(8^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{8^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{8^{1}}$

Evaluate the exponent.

$\frac{\sqrt{10} {\sqrt[4]{8}}^{3}}{8}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[4]{8}}^{3}$ as $\sqrt[4]{8^{3}}$ .

$\frac{\sqrt{10} \sqrt[4]{8^{3}}}{8}$

Raise $8$ to the power of $3$ .

$\frac{\sqrt{10} \sqrt[4]{512}}{8}$

Rewrite $512$ as $4^{4} \cdot 2$ .

Factor $256$ out of $512$ .

$\frac{\sqrt{10} \sqrt[4]{256 (2)}}{8}$

Rewrite $256$ as $4^{4}$ .

$\frac{\sqrt{10} \sqrt[4]{4^{4} \cdot 2}}{8}$

Pull terms out from under the radical.

$\frac{\sqrt{10} \cdot 4 \sqrt[4]{2}}{8}$

Combine exponents.

Rewrite the expression using the least common index of $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{4 (10^{\frac{1}{2}} \sqrt[4]{2})}{8}$

Rewrite $10^{\frac{1}{2}}$ as $10^{\frac{2}{4}}$ .

$\frac{4 (10^{\frac{2}{4}} \sqrt[4]{2})}{8}$

Rewrite $10^{\frac{2}{4}}$ as $\sqrt[4]{10^{2}}$ .

$\frac{4 (\sqrt[4]{10^{2}} \sqrt[4]{2})}{8}$

Combine using the product rule for radicals.

$\frac{4 \sqrt[4]{10^{2} \cdot 2}}{8}$

Raise $10$ to the power of $2$ .

$\frac{4 \sqrt[4]{100 \cdot 2}}{8}$

Multiply $100$ by $2$ .

$\frac{4 \sqrt[4]{200}}{8}$

Step 4

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 \sqrt[4]{200}$ .

$\frac{4 (\sqrt[4]{200})}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$\frac{4 \sqrt[4]{200}}{4 \cdot 2}$

Cancel the common factor.

$\frac{4 \sqrt[4]{200}}{4 \cdot 2}$

Rewrite the expression.

$\frac{\sqrt[4]{200}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{200}}{2}$

Decimal Form:

$1.88030154 \dots$

Simplify ( square root of 10)/( fourth root of 8)

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