Evaluate 4 square root of 10*(2 square root of 90)

$4 \sqrt{10} \cdot (2 \sqrt{90})$

Step 1

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$4 \sqrt{10} \cdot (2 \sqrt{9 (10)})$

Rewrite $9$ as $3^{2}$ .

$4 \sqrt{10} \cdot (2 \sqrt{3^{2} \cdot 10})$

Step 2

Pull terms out from under the radical.

$4 \sqrt{10} \cdot (2 (3 \sqrt{10}))$

Step 3

Multiply $3$ by $2$ .

$4 \sqrt{10} \cdot (6 \sqrt{10})$

Step 4

Multiply $4 \sqrt{10} (6 \sqrt{10})$ .

Multiply $6$ by $4$ .

$24 \sqrt{10} \sqrt{10}$

Raise $\sqrt{10}$ to the power of $1$ .

$24 ({\sqrt{10}}^{1} \sqrt{10})$

Raise $\sqrt{10}$ to the power of $1$ .

$24 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

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

$24 {\sqrt{10}}^{1 + 1}$

Add $1$ and $1$ .

$24 {\sqrt{10}}^{2}$

Step 5

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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

$24 {(10^{\frac{1}{2}})}^{2}$

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

$24 \cdot 10^{\frac{1}{2} \cdot 2}$

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

$24 \cdot 10^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$24 \cdot 10^{\frac{2}{2}}$

Rewrite the expression.

$24 \cdot 10^{1}$

Evaluate the exponent.

$24 \cdot 10$

Step 6

Multiply $24$ by $10$ .

$240$