Pre-Algebra Examples

$4 x \sqrt{45 x^{7}} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $9$ out of $45$ .

$4 x \sqrt{9 (5) x^{7}} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.2

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

$4 x \sqrt{3^{2} \cdot 5 x^{7}} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.3

Factor out $x^{6}$ .

$4 x \sqrt{3^{2} \cdot 5 (x^{6} x)} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.4

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

$4 x \sqrt{3^{2} \cdot 5 ({(x^{3})}^{2} x)} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.5

Move $5$ .

$4 x \sqrt{3^{2} {(x^{3})}^{2} \cdot 5 x} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.6

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

$4 x \sqrt{{(3 x^{3})}^{2} \cdot 5 x} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.1.7

Add parentheses.

$4 x \sqrt{{(3 x^{3})}^{2} \cdot (5 x)} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.2

Pull terms out from under the radical.

$4 x (3 x^{3} \sqrt{5 x}) + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.3

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

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Step 1.3.1

Move $x^{3}$ .

$4 (x^{3} x) (3 \sqrt{5 x}) + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.3.2

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

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Step 1.3.2.1

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

$4 (x^{3} x^{1}) (3 \sqrt{5 x}) + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.3.2.2

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

$4 x^{3 + 1} (3 \sqrt{5 x}) + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.3.3

Add $3$ and $1$ .

$4 x^{4} (3 \sqrt{5 x}) + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.4

Multiply $3$ by $4$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{20 x^{5}}$

Step 1.5

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

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Step 1.5.1

Factor $4$ out of $20$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{4 (5) x^{5}}$

Step 1.5.2

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

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{2^{2} \cdot 5 x^{5}}$

Step 1.5.3

Factor out $x^{4}$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{2^{2} \cdot 5 (x^{4} x)}$

Step 1.5.4

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{2^{2} \cdot 5 ({(x^{2})}^{2} x)}$

Step 1.5.5

Move $5$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{2^{2} {(x^{2})}^{2} \cdot 5 x}$

Step 1.5.6

Rewrite $2^{2} {(x^{2})}^{2}$ as ${(2 x^{2})}^{2}$ .

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{{(2 x^{2})}^{2} \cdot 5 x}$

Step 1.5.7

Add parentheses.

$12 x^{4} \sqrt{5 x} + 6 x^{2} \sqrt{{(2 x^{2})}^{2} \cdot (5 x)}$

Step 1.6

Pull terms out from under the radical.

$12 x^{4} \sqrt{5 x} + 6 x^{2} (2 x^{2} \sqrt{5 x})$

Step 1.7

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.7.1

Move $x^{2}$ .

$12 x^{4} \sqrt{5 x} + 6 (x^{2} x^{2}) (2 \sqrt{5 x})$

Step 1.7.2

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

$12 x^{4} \sqrt{5 x} + 6 x^{2 + 2} (2 \sqrt{5 x})$

Step 1.7.3

Add $2$ and $2$ .

$12 x^{4} \sqrt{5 x} + 6 x^{4} (2 \sqrt{5 x})$

Step 1.8

Multiply $2$ by $6$ .

$12 x^{4} \sqrt{5 x} + 12 x^{4} \sqrt{5 x}$

Step 2

Add $12 x^{4} \sqrt{5 x}$ and $12 x^{4} \sqrt{5 x}$ .

$24 x^{4} \sqrt{5 x}$