Simplify the Radical Expression (3 square root of 8)/(8 square root of 5)

$\frac{3 \sqrt{8}}{8 \sqrt{5}}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{3 \sqrt{4 (2)}}{8 \sqrt{5}}$

Rewrite $4$ as $2^{2}$ .

$\frac{3 \sqrt{2^{2} \cdot 2}}{8 \sqrt{5}}$

Pull terms out from under the radical.

$\frac{3 (| 2 | \sqrt{2})}{8 \sqrt{5}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{3 \cdot 2 \sqrt{2}}{8 \sqrt{5}}$

Multiply $3$ by $2$ .

$\frac{6 \sqrt{2}}{8 \sqrt{5}}$

Step 2

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{2 (3 \sqrt{2})}{8 \sqrt{5}}$

Cancel the common factors.

Factor $2$ out of $8 \sqrt{5}$ .

$\frac{2 (3 \sqrt{2})}{2 (4 \sqrt{5})}$

Cancel the common factor.

$\frac{2 (3 \sqrt{2})}{2 (4 \sqrt{5})}$

Rewrite the expression.

$\frac{3 \sqrt{2}}{4 \sqrt{5}}$

Step 3

Multiply $\frac{3 \sqrt{2}}{4 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2}}{4 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{3 \sqrt{2}}{4 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 \sqrt{5} \sqrt{5}}$

Move $\sqrt{5}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 (\sqrt{5} \sqrt{5})}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 ({\sqrt{5}}^{1} \sqrt{5})}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

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

$\frac{3 \sqrt{2} \sqrt{5}}{4 {\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 {\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 {(5^{\frac{1}{2}})}^{2}}$

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

$\frac{3 \sqrt{2} \sqrt{5}}{4 \cdot 5^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \sqrt{2} \sqrt{5}}{4 \cdot 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \sqrt{2} \sqrt{5}}{4 \cdot 5^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{3 \sqrt{2} \sqrt{5}}{4 \cdot 5^{1}}$

Evaluate the exponent.

$\frac{3 \sqrt{2} \sqrt{5}}{4 \cdot 5}$

Step 5

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{3 \sqrt{5 \cdot 2}}{4 \cdot 5}$

Multiply $5$ by $2$ .

$\frac{3 \sqrt{10}}{4 \cdot 5}$

Step 6

Multiply $4$ by $5$ .

$\frac{3 \sqrt{10}}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{20}$

Decimal Form:

$0.47434164 \dots$

Simplify the Radical Expression (3 square root of 8)/(8 square root of 5)

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