Algebra Examples

$\frac{4}{5} \sqrt{72 y^{7}} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $72 y^{7}$ as ${(6 y^{3})}^{2} \cdot (2 y)$ .

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Step 1.1.1

Factor $36$ out of $72$ .

$\frac{4}{5} \cdot \sqrt{36 (2) y^{7}} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.2

Rewrite $36$ as $6^{2}$ .

$\frac{4}{5} \cdot \sqrt{6^{2} \cdot 2 y^{7}} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.3

Factor out $y^{6}$ .

$\frac{4}{5} \cdot \sqrt{6^{2} \cdot 2 (y^{6} y)} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.4

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

$\frac{4}{5} \cdot \sqrt{6^{2} \cdot 2 ({(y^{3})}^{2} y)} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.5

Move $2$ .

$\frac{4}{5} \cdot \sqrt{6^{2} {(y^{3})}^{2} \cdot 2 y} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.6

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

$\frac{4}{5} \cdot \sqrt{{(6 y^{3})}^{2} \cdot 2 y} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.1.7

Add parentheses.

$\frac{4}{5} \cdot \sqrt{{(6 y^{3})}^{2} \cdot (2 y)} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.2

Pull terms out from under the radical.

$\frac{4}{5} \cdot (6 y^{3} \sqrt{2 y}) - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.3

Multiply $\frac{4}{5} (6 y^{3} \sqrt{2 y})$ .

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Step 1.3.1

Combine $6$ and $\frac{4}{5}$ .

$\frac{6 \cdot 4}{5} (y^{3} \sqrt{2 y}) - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.3.2

Multiply $6$ by $4$ .

$\frac{24}{5} (y^{3} \sqrt{2 y}) - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.3.3

Combine $y^{3}$ and $\frac{24}{5}$ .

$\frac{y^{3} \cdot 24}{5} \sqrt{2 y} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.3.4

Combine $\frac{y^{3} \cdot 24}{5}$ and $\sqrt{2 y}$ .

$\frac{y^{3} \cdot 24 \sqrt{2 y}}{5} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.4

Move $24$ to the left of $y^{3}$ .

$\frac{24 \cdot y^{3} \sqrt{2 y}}{5} - \sqrt{128 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.5

Rewrite $128 y^{7}$ as ${(8 y^{3})}^{2} \cdot (2 y)$ .

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Step 1.5.1

Factor $64$ out of $128$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{64 (2) y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.5.2

Rewrite $64$ as $8^{2}$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{8^{2} \cdot 2 y^{7}} + 5 y^{3} \sqrt{18 y}$

Step 1.5.3

Factor out $y^{6}$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{8^{2} \cdot 2 (y^{6} y)} + 5 y^{3} \sqrt{18 y}$

Step 1.5.4

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

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{8^{2} \cdot 2 ({(y^{3})}^{2} y)} + 5 y^{3} \sqrt{18 y}$

Step 1.5.5

Move $2$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{8^{2} {(y^{3})}^{2} \cdot 2 y} + 5 y^{3} \sqrt{18 y}$

Step 1.5.6

Rewrite $8^{2} {(y^{3})}^{2}$ as ${(8 y^{3})}^{2}$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{{(8 y^{3})}^{2} \cdot 2 y} + 5 y^{3} \sqrt{18 y}$

Step 1.5.7

Add parentheses.

$\frac{24 y^{3} \sqrt{2 y}}{5} - \sqrt{{(8 y^{3})}^{2} \cdot (2 y)} + 5 y^{3} \sqrt{18 y}$

Step 1.6

Pull terms out from under the radical.

$\frac{24 y^{3} \sqrt{2 y}}{5} - (8 y^{3} \sqrt{2 y}) + 5 y^{3} \sqrt{18 y}$

Step 1.7

Multiply $8$ by $- 1$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 (y^{3} \sqrt{2 y}) + 5 y^{3} \sqrt{18 y}$

Step 1.8

Rewrite $18 y$ as $3^{2} \cdot (2 y)$ .

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Step 1.8.1

Factor $9$ out of $18$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} + 5 y^{3} \sqrt{9 (2) y}$

Step 1.8.2

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

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} + 5 y^{3} \sqrt{3^{2} \cdot 2 y}$

Step 1.8.3

Add parentheses.

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} + 5 y^{3} \sqrt{3^{2} \cdot (2 y)}$

Step 1.9

Pull terms out from under the radical.

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} + 5 y^{3} (3 \sqrt{2 y})$

Step 1.10

Multiply $3$ by $5$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} + 15 y^{3} \sqrt{2 y}$

Step 2

To write $- 8 y^{3} \sqrt{2 y}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} - 8 y^{3} \sqrt{2 y} \cdot \frac{5}{5} + 15 y^{3} \sqrt{2 y}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 8 y^{3} \sqrt{2 y}$ and $\frac{5}{5}$ .

$\frac{24 y^{3} \sqrt{2 y}}{5} + \frac{- 8 y^{3} \sqrt{2 y} \cdot 5}{5} + 15 y^{3} \sqrt{2 y}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{24 y^{3} \sqrt{2 y} - 8 y^{3} \sqrt{2 y} \cdot 5}{5} + 15 y^{3} \sqrt{2 y}$

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $8 y^{3} \sqrt{2 y}$ out of $24 y^{3} \sqrt{2 y} - 8 y^{3} \sqrt{2 y} \cdot 5$ .

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Step 4.1.1.1

Factor $8 y^{3} \sqrt{2 y}$ out of $24 y^{3} \sqrt{2 y}$ .

$\frac{8 y^{3} \sqrt{2 y} (3) - 8 y^{3} \sqrt{2 y} \cdot 5}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.1.1.2

Factor $8 y^{3} \sqrt{2 y}$ out of $- 8 y^{3} \sqrt{2 y} \cdot 5$ .

$\frac{8 y^{3} \sqrt{2 y} (3) + 8 y^{3} \sqrt{2 y} (- 1 \cdot 5)}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.1.1.3

Factor $8 y^{3} \sqrt{2 y}$ out of $8 y^{3} \sqrt{2 y} (3) + 8 y^{3} \sqrt{2 y} (- 1 \cdot 5)$ .

$\frac{8 y^{3} \sqrt{2 y} (3 - 1 \cdot 5)}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.1.2

Multiply $- 1$ by $5$ .

$\frac{8 y^{3} \sqrt{2 y} (3 - 5)}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.1.3

Subtract $5$ from $3$ .

$\frac{8 y^{3} \sqrt{2 y} \cdot - 2}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.1.4

Multiply $- 2$ by $8$ .

$\frac{- 16 y^{3} \sqrt{2 y}}{5} + 15 y^{3} \sqrt{2 y}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{16 y^{3} \sqrt{2 y}}{5} + 15 y^{3} \sqrt{2 y}$

Step 5

To write $15 y^{3} \sqrt{2 y}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{16 y^{3} \sqrt{2 y}}{5} + 15 y^{3} \sqrt{2 y} \cdot \frac{5}{5}$

Step 6

Simplify terms.

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Step 6.1

Combine $15 y^{3} \sqrt{2 y}$ and $\frac{5}{5}$ .

$- \frac{16 y^{3} \sqrt{2 y}}{5} + \frac{15 y^{3} \sqrt{2 y} \cdot 5}{5}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{- 16 y^{3} \sqrt{2 y} + 15 y^{3} \sqrt{2 y} \cdot 5}{5}$

Step 7

Simplify the numerator.

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Step 7.1

Factor $y^{3} \sqrt{2 y}$ out of $- 16 y^{3} \sqrt{2 y} + 15 y^{3} \sqrt{2 y} \cdot 5$ .

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Step 7.1.1

Factor $y^{3} \sqrt{2 y}$ out of $- 16 y^{3} \sqrt{2 y}$ .

$\frac{y^{3} \sqrt{2 y} (- 16) + 15 y^{3} \sqrt{2 y} \cdot 5}{5}$

Step 7.1.2

Factor $y^{3} \sqrt{2 y}$ out of $15 y^{3} \sqrt{2 y} \cdot 5$ .

$\frac{y^{3} \sqrt{2 y} (- 16) + y^{3} \sqrt{2 y} (15 \cdot 5)}{5}$

Step 7.1.3

Factor $y^{3} \sqrt{2 y}$ out of $y^{3} \sqrt{2 y} (- 16) + y^{3} \sqrt{2 y} (15 \cdot 5)$ .

$\frac{y^{3} \sqrt{2 y} (- 16 + 15 \cdot 5)}{5}$

Step 7.2

Multiply $15$ by $5$ .

$\frac{y^{3} \sqrt{2 y} (- 16 + 75)}{5}$

Step 7.3

Add $- 16$ and $75$ .

$\frac{y^{3} \sqrt{2 y} \cdot 59}{5}$

Step 8

Move $59$ to the left of $y^{3} \sqrt{2 y}$ .

$\frac{59 y^{3} \sqrt{2 y}}{5}$