Simplify (20 square root of z^6)÷( square root of 16z^7)

$20 \sqrt{z^{6}} \div \sqrt{16 z^{7}}$

Step 1

Rewrite the division as a fraction.

$\frac{20 \sqrt{z^{6}}}{\sqrt{16 z^{7}}}$

Step 2

Combine $\sqrt{z^{6}}$ and $\sqrt{16 z^{7}}$ into a single radical.

$20 \sqrt{\frac{z^{6}}{16 z^{7}}}$

Step 3

Reduce the expression $\frac{z^{6}}{16 z^{7}}$ by cancelling the common factors.

Multiply by $1$ .

$20 \sqrt{\frac{z^{6} \cdot 1}{16 z^{7}}}$

Factor $z^{6}$ out of $16 z^{7}$ .

$20 \sqrt{\frac{z^{6} \cdot 1}{z^{6} (16 z)}}$

Cancel the common factor.

$20 \sqrt{\frac{z^{6} \cdot 1}{z^{6} (16 z)}}$

Rewrite the expression.

$20 \sqrt{\frac{1}{16 z}}$

Step 4

Rewrite $\frac{1}{16 z}$ as ${(\frac{1}{2^{2}})}^{2} \frac{1}{z}$ .

Factor the perfect power $1^{2}$ out of $1$ .

$20 \sqrt{\frac{1^{2} \cdot 1}{16 z}}$

Factor the perfect power ${(2^{2})}^{2}$ out of $16 z$ .

$20 \sqrt{\frac{1^{2} \cdot 1}{{(2^{2})}^{2} z}}$

Rearrange the fraction $\frac{1^{2} \cdot 1}{{(2^{2})}^{2} z}$ .

$20 \sqrt{{(\frac{1}{2^{2}})}^{2} \frac{1}{z}}$

Step 5

Pull terms out from under the radical.

$20 (\frac{1}{2^{2}} \sqrt{\frac{1}{z}})$

Step 6

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

$20 (\frac{1}{4} \sqrt{\frac{1}{z}})$

Step 7

Rewrite $\sqrt{\frac{1}{z}}$ as $\frac{\sqrt{1}}{\sqrt{z}}$ .

$20 (\frac{1}{4} \cdot \frac{\sqrt{1}}{\sqrt{z}})$

Step 8

Any root of $1$ is $1$ .

$20 (\frac{1}{4} \cdot \frac{1}{\sqrt{z}})$

Step 9

Multiply $\frac{1}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$20 (\frac{1}{4} (\frac{1}{\sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}}))$

Step 10

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{\sqrt{z} \sqrt{z}})$

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{{\sqrt{z}}^{1} \sqrt{z}})$

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{{\sqrt{z}}^{1} {\sqrt{z}}^{1}})$

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{{\sqrt{z}}^{1 + 1}})$

Add $1$ and $1$ .

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{{\sqrt{z}}^{2}})$

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

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{{(z^{\frac{1}{2}})}^{2}})$

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{z^{\frac{1}{2} \cdot 2}})$

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

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{z^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{z^{\frac{2}{2}}})$

Rewrite the expression.

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{z^{1}})$

Simplify.

$20 (\frac{1}{4} \cdot \frac{\sqrt{z}}{z})$

Step 11

Multiply $\frac{1}{4}$ by $\frac{\sqrt{z}}{z}$ .

$20 \frac{\sqrt{z}}{4 z}$

Step 12

Cancel the common factor of $4$ .

Factor $4$ out of $20$ .

$4 (5) \frac{\sqrt{z}}{4 z}$

Factor $4$ out of $4 z$ .

$4 (5) \frac{\sqrt{z}}{4 (z)}$

Cancel the common factor.

$4 \cdot 5 \frac{\sqrt{z}}{4 z}$

Rewrite the expression.

$5 \frac{\sqrt{z}}{z}$

Step 13

Combine $5$ and $\frac{\sqrt{z}}{z}$ .

$\frac{5 \sqrt{z}}{z}$

Simplify (20 square root of z^6)÷( square root of 16z^7)

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