Algebra Examples

$\sqrt{5 a^{7}} \sqrt{15 a^{3}}$

Step 1

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

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Step 1.1

Factor out $a^{6}$ .

$\sqrt{5 (a^{6} a)} \sqrt{15 a^{3}}$

Step 1.2

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

$\sqrt{5 ({(a^{3})}^{2} a)} \sqrt{15 a^{3}}$

Step 1.3

Reorder $5$ and ${(a^{3})}^{2}$ .

$\sqrt{{(a^{3})}^{2} \cdot 5 a} \sqrt{15 a^{3}}$

Step 1.4

Add parentheses.

$\sqrt{{(a^{3})}^{2} \cdot (5 a)} \sqrt{15 a^{3}}$

Step 2

Pull terms out from under the radical.

$a^{3} \sqrt{5 a} \sqrt{15 a^{3}}$

Step 3

Rewrite $15 a^{3}$ as $a^{2} \cdot (15 a)$ .

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Step 3.1

Factor out $a^{2}$ .

$a^{3} \sqrt{5 a} \sqrt{15 (a^{2} a)}$

Step 3.2

Reorder $15$ and $a^{2}$ .

$a^{3} \sqrt{5 a} \sqrt{a^{2} \cdot 15 a}$

Step 3.3

Add parentheses.

$a^{3} \sqrt{5 a} \sqrt{a^{2} \cdot (15 a)}$

Step 4

Pull terms out from under the radical.

$a^{3} \sqrt{5 a} (a \sqrt{15 a})$

Step 5

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

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Step 5.1

Move $a$ .

$a \cdot a^{3} \sqrt{5 a} \sqrt{15 a}$

Step 5.2

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

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Step 5.2.1

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

$a^{1} a^{3} \sqrt{5 a} \sqrt{15 a}$

Step 5.2.2

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

$a^{1 + 3} \sqrt{5 a} \sqrt{15 a}$

Step 5.3

Add $1$ and $3$ .

$a^{4} \sqrt{5 a} \sqrt{15 a}$

Step 6

Multiply $a^{4} \sqrt{5 a} \sqrt{15 a}$ .

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Step 6.1

Combine using the product rule for radicals.

$a^{4} \sqrt{15 a (5 a)}$

Step 6.2

Multiply $5$ by $15$ .

$a^{4} \sqrt{75 a \cdot a}$

Step 6.3

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

$a^{4} \sqrt{75 (a^{1} a)}$

Step 6.4

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

$a^{4} \sqrt{75 (a^{1} a^{1})}$

Step 6.5

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

$a^{4} \sqrt{75 a^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$a^{4} \sqrt{75 a^{2}}$

Step 7

Rewrite $75 a^{2}$ as ${(5 a)}^{2} \cdot 3$ .

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Step 7.1

Factor $25$ out of $75$ .

$a^{4} \sqrt{25 (3) a^{2}}$

Step 7.2

Rewrite $25$ as $5^{2}$ .

$a^{4} \sqrt{5^{2} \cdot 3 a^{2}}$

Step 7.3

Move $3$ .

$a^{4} \sqrt{5^{2} a^{2} \cdot 3}$

Step 7.4

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

$a^{4} \sqrt{{(5 a)}^{2} \cdot 3}$

Step 8

Pull terms out from under the radical.

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

Step 9

Rewrite using the commutative property of multiplication.

$5 \sqrt{3} a^{4} a$

Step 10

Multiply $a^{4}$ by $a$ by adding the exponents.

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Step 10.1

Move $a$ .

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

Step 10.2

Multiply $a$ by $a^{4}$ .

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Step 10.2.1

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

$5 \sqrt{3} (a^{1} a^{4})$

Step 10.2.2

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

$5 \sqrt{3} a^{1 + 4}$

Step 10.3

Add $1$ and $4$ .

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