Simplify the Radical Expression -4 square root of 28x* square root of 7x^3

$- 4 \sqrt{28 x} \cdot \sqrt{7 x^{3}}$

Step 1

Rewrite $28 x$ as $2^{2} \cdot (7 x)$ .

Factor $4$ out of $28$ .

$- 4 \sqrt{4 (7) x} \cdot \sqrt{7 x^{3}}$

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

$- 4 \sqrt{2^{2} \cdot 7 x} \cdot \sqrt{7 x^{3}}$

Add parentheses.

$- 4 \sqrt{2^{2} \cdot (7 x)} \cdot \sqrt{7 x^{3}}$

Step 2

Pull terms out from under the radical.

$- 4 (| 2 | \sqrt{7 x}) \cdot \sqrt{7 x^{3}}$

Step 3

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

$- 4 (2 \sqrt{7 x}) \cdot \sqrt{7 x^{3}}$

Step 4

Multiply $2$ by $- 4$ .

$- 8 \sqrt{7 x} \cdot \sqrt{7 x^{3}}$

Step 5

Rewrite $7 x^{3}$ as $x^{2} \cdot (7 x)$ .

Factor out $x^{2}$ .

$- 8 \sqrt{7 x} \cdot \sqrt{7 (x^{2} x)}$

Reorder $7$ and $x^{2}$ .

$- 8 \sqrt{7 x} \cdot \sqrt{x^{2} \cdot 7 x}$

Add parentheses.

$- 8 \sqrt{7 x} \cdot \sqrt{x^{2} \cdot (7 x)}$

Step 6

Pull terms out from under the radical.

$- 8 \sqrt{7 x} \cdot (| x | \sqrt{7 x})$

Step 7

Multiply $- 8 \sqrt{7 x} (| x | \sqrt{7 x})$ .

Raise $\sqrt{7 x}$ to the power of $1$ .

$- 8 ({\sqrt{7 x}}^{1} \sqrt{7 x}) | x |$

Raise $\sqrt{7 x}$ to the power of $1$ .

$- 8 ({\sqrt{7 x}}^{1} {\sqrt{7 x}}^{1}) | x |$

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

$- 8 {\sqrt{7 x}}^{1 + 1} | x |$

Add $1$ and $1$ .

$- 8 {\sqrt{7 x}}^{2} | x |$

Step 8

Rewrite ${\sqrt{7 x}}^{2}$ as $7 x$ .

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

$- 8 {({(7 x)}^{\frac{1}{2}})}^{2} | x |$

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

$- 8 {(7 x)}^{\frac{1}{2} \cdot 2} | x |$

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

$- 8 {(7 x)}^{\frac{2}{2}} | x |$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 8 {(7 x)}^{\frac{2}{2}} | x |$

Rewrite the expression.

$- 8 {(7 x)}^{1} | x |$

Simplify.

$- 8 (7 x) | x |$

Step 9

Multiply $7$ by $- 8$ .

$- 56 x | x |$

Simplify the Radical Expression -4 square root of 28x* square root of 7x^3

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