Simplify the Radical Expression ( square root of 5a)( square root of 2a)(3 square root of 10a^2)

$(\sqrt{5 a}) (\sqrt{2 a}) (3 \sqrt{10 a^{2}})$

Step 1

Rewrite using the commutative property of multiplication.

$3 \sqrt{5 a} \sqrt{2 a} \sqrt{10 a^{2}}$

Step 2

Multiply $3 \sqrt{5 a} \sqrt{2 a}$ .

Combine using the product rule for radicals.

$3 \sqrt{2 a (5 a)} \sqrt{10 a^{2}}$

Multiply $5$ by $2$ .

$3 \sqrt{10 a \cdot a} \sqrt{10 a^{2}}$

Raise $a$ to the power of $1$ .

$3 \sqrt{10 (a^{1} a)} \sqrt{10 a^{2}}$

Raise $a$ to the power of $1$ .

$3 \sqrt{10 (a^{1} a^{1})} \sqrt{10 a^{2}}$

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

$3 \sqrt{10 a^{1 + 1}} \sqrt{10 a^{2}}$

Add $1$ and $1$ .

$3 \sqrt{10 a^{2}} \sqrt{10 a^{2}}$

Step 3

Reorder $10$ and $a^{2}$ .

$3 \sqrt{a^{2} \cdot 10} \sqrt{10 a^{2}}$

Step 4

Pull terms out from under the radical.

$3 (| a | \sqrt{10}) \sqrt{10 a^{2}}$

Step 5

Reorder $10$ and $a^{2}$ .

$3 | a | \sqrt{10} \sqrt{a^{2} \cdot 10}$

Step 6

Pull terms out from under the radical.

$3 | a | \sqrt{10} (| a | \sqrt{10})$

Step 7

Multiply $3 | a | \sqrt{10} (| a | \sqrt{10})$ .

To multiply absolute values, multiply the terms inside each absolute value.

$3 | a \cdot a | \sqrt{10} \sqrt{10}$

Raise $a$ to the power of $1$ .

$3 | a^{1} a | \sqrt{10} \sqrt{10}$

Raise $a$ to the power of $1$ .

$3 | a^{1} a^{1} | \sqrt{10} \sqrt{10}$

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

$3 | a^{1 + 1} | \sqrt{10} \sqrt{10}$

Add $1$ and $1$ .

$3 | a^{2} | \sqrt{10} \sqrt{10}$

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

$3 | a^{2} | ({\sqrt{10}}^{1} \sqrt{10})$

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

$3 | a^{2} | ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

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

$3 | a^{2} | {\sqrt{10}}^{1 + 1}$

Add $1$ and $1$ .

$3 | a^{2} | {\sqrt{10}}^{2}$

Step 8

Remove non-negative terms from the absolute value.

$3 a^{2} {\sqrt{10}}^{2}$

Step 9

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

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

$3 a^{2} {(10^{\frac{1}{2}})}^{2}$

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

$3 a^{2} \cdot 10^{\frac{1}{2} \cdot 2}$

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

$3 a^{2} \cdot 10^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 a^{2} \cdot 10^{\frac{2}{2}}$

Rewrite the expression.

$3 a^{2} \cdot 10^{1}$

Evaluate the exponent.

$3 a^{2} \cdot 10$

Step 10

Multiply $10$ by $3$ .

$30 a^{2}$

Simplify the Radical Expression ( square root of 5a)( square root of 2a)(3 square root of 10a^2)

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