Pre-Algebra Examples

$\sqrt{45 a^{7}} \cdot \sqrt{20 a}$

Step 1

Rewrite $45 a^{7}$ as ${(3 a^{3})}^{2} \cdot (5 a)$ .

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Step 1.1

Factor $9$ out of $45$ .

$\sqrt{9 (5) a^{7}} \cdot \sqrt{20 a}$

Step 1.2

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

$\sqrt{3^{2} \cdot 5 a^{7}} \cdot \sqrt{20 a}$

Step 1.3

Factor out $a^{6}$ .

$\sqrt{3^{2} \cdot 5 (a^{6} a)} \cdot \sqrt{20 a}$

Step 1.4

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

$\sqrt{3^{2} \cdot 5 ({(a^{3})}^{2} a)} \cdot \sqrt{20 a}$

Step 1.5

Move $5$ .

$\sqrt{3^{2} {(a^{3})}^{2} \cdot 5 a} \cdot \sqrt{20 a}$

Step 1.6

Rewrite $3^{2} {(a^{3})}^{2}$ as ${(3 a^{3})}^{2}$ .

$\sqrt{{(3 a^{3})}^{2} \cdot 5 a} \cdot \sqrt{20 a}$

Step 1.7

Add parentheses.

$\sqrt{{(3 a^{3})}^{2} \cdot (5 a)} \cdot \sqrt{20 a}$

Step 2

Pull terms out from under the radical.

$3 a^{3} \sqrt{5 a} \cdot \sqrt{20 a}$

Step 3

Rewrite $20 a$ as $2^{2} \cdot (5 a)$ .

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Step 3.1

Factor $4$ out of $20$ .

$3 a^{3} \sqrt{5 a} \cdot \sqrt{4 (5) a}$

Step 3.2

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

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

Step 3.3

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Multiply $2$ by $3$ .

$6 a^{3} \sqrt{5 a} \sqrt{5 a}$

Step 5.2

Raise $\sqrt{5 a}$ to the power of $1$ .

$6 a^{3} ({\sqrt{5 a}}^{1} \sqrt{5 a})$

Step 5.3

Raise $\sqrt{5 a}$ to the power of $1$ .

$6 a^{3} ({\sqrt{5 a}}^{1} {\sqrt{5 a}}^{1})$

Step 5.4

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

$6 a^{3} {\sqrt{5 a}}^{1 + 1}$

Step 5.5

Add $1$ and $1$ .

$6 a^{3} {\sqrt{5 a}}^{2}$

Step 6

Rewrite ${\sqrt{5 a}}^{2}$ as $5 a$ .

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a}$ as ${(5 a)}^{\frac{1}{2}}$ .

$6 a^{3} {({(5 a)}^{\frac{1}{2}})}^{2}$

Step 6.2

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

$6 a^{3} {(5 a)}^{\frac{1}{2} \cdot 2}$

Step 6.3

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

$6 a^{3} {(5 a)}^{\frac{2}{2}}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$6 a^{3} {(5 a)}^{\frac{2}{2}}$

Step 6.4.2

Rewrite the expression.

$6 a^{3} {(5 a)}^{1}$

Step 6.5

Simplify.

$6 a^{3} (5 a)$

Step 7

Rewrite using the commutative property of multiplication.

$6 \cdot 5 a^{3} a$

Step 8

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 8.1

Move $a$ .

$6 \cdot 5 (a \cdot a^{3})$

Step 8.2

Multiply $a$ by $a^{3}$ .

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Step 8.2.1

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

$6 \cdot 5 (a^{1} a^{3})$

Step 8.2.2

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

$6 \cdot 5 a^{1 + 3}$

Step 8.3

Add $1$ and $3$ .

$6 \cdot 5 a^{4}$

Step 9

Multiply $6$ by $5$ .

$30 a^{4}$