Simplify (6d^3)/( square root of 8d^3)

$\frac{6 d^{3}}{\sqrt{8 d^{3}}}$

Step 1

Simplify the denominator.

Rewrite $8 d^{3}$ as ${(2 d)}^{2} \cdot (2 d)$ .

Factor $4$ out of $8$ .

$\frac{6 d^{3}}{\sqrt{4 (2) d^{3}}}$

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

$\frac{6 d^{3}}{\sqrt{2^{2} \cdot 2 d^{3}}}$

Factor out $d^{2}$ .

$\frac{6 d^{3}}{\sqrt{2^{2} \cdot 2 (d^{2} d)}}$

Move $2$ .

$\frac{6 d^{3}}{\sqrt{2^{2} d^{2} \cdot 2 d}}$

Rewrite $2^{2} d^{2}$ as ${(2 d)}^{2}$ .

$\frac{6 d^{3}}{\sqrt{{(2 d)}^{2} \cdot 2 d}}$

Add parentheses.

$\frac{6 d^{3}}{\sqrt{{(2 d)}^{2} \cdot (2 d)}}$

Pull terms out from under the radical.

$\frac{6 d^{3}}{2 d \sqrt{2 d}}$

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 d^{3}$ .

$\frac{2 (3 d^{3})}{2 d \sqrt{2 d}}$

Cancel the common factors.

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

$\frac{2 (3 d^{3})}{2 (d \sqrt{2 d})}$

Cancel the common factor.

$\frac{2 (3 d^{3})}{2 (d \sqrt{2 d})}$

Rewrite the expression.

$\frac{3 d^{3}}{d \sqrt{2 d}}$

Cancel the common factor of $d^{3}$ and $d$ .

Factor $d$ out of $3 d^{3}$ .

$\frac{d (3 d^{2})}{d \sqrt{2 d}}$

Cancel the common factors.

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

$\frac{d (3 d^{2})}{d (\sqrt{2 d})}$

Cancel the common factor.

$\frac{d (3 d^{2})}{d \sqrt{2 d}}$

Rewrite the expression.

$\frac{3 d^{2}}{\sqrt{2 d}}$

Step 3

Multiply $\frac{3 d^{2}}{\sqrt{2 d}}$ by $\frac{\sqrt{2 d}}{\sqrt{2 d}}$ .

$\frac{3 d^{2}}{\sqrt{2 d}} \cdot \frac{\sqrt{2 d}}{\sqrt{2 d}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{3 d^{2}}{\sqrt{2 d}}$ by $\frac{\sqrt{2 d}}{\sqrt{2 d}}$ .

$\frac{3 d^{2} \sqrt{2 d}}{\sqrt{2 d} \sqrt{2 d}}$

Raise $\sqrt{2 d}$ to the power of $1$ .

$\frac{3 d^{2} \sqrt{2 d}}{{\sqrt{2 d}}^{1} \sqrt{2 d}}$

Raise $\sqrt{2 d}$ to the power of $1$ .

$\frac{3 d^{2} \sqrt{2 d}}{{\sqrt{2 d}}^{1} {\sqrt{2 d}}^{1}}$

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

$\frac{3 d^{2} \sqrt{2 d}}{{\sqrt{2 d}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 d^{2} \sqrt{2 d}}{{\sqrt{2 d}}^{2}}$

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

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

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

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

$\frac{3 d^{2} \sqrt{2 d}}{{(2 d)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{3 d^{2} \sqrt{2 d}}{{(2 d)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 d^{2} \sqrt{2 d}}{{(2 d)}^{\frac{2}{2}}}$

Rewrite the expression.

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

Simplify.

$\frac{3 d^{2} \sqrt{2 d}}{2 d}$

Step 5

Cancel the common factor of $d^{2}$ and $d$ .

Factor $d$ out of $3 d^{2} \sqrt{2 d}$ .

$\frac{d (3 d \sqrt{2 d})}{2 d}$

Cancel the common factors.

Factor $d$ out of $2 d$ .

$\frac{d (3 d \sqrt{2 d})}{d \cdot 2}$

Cancel the common factor.

$\frac{d (3 d \sqrt{2 d})}{d \cdot 2}$

Rewrite the expression.

$\frac{3 d \sqrt{2 d}}{2}$

Simplify (6d^3)/( square root of 8d^3)

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