Basic Math Examples

$20 \sqrt{12} a^{5} y^{7} - a y^{2} \sqrt{3 a^{5} y^{3}}$

Step 1

Factor $a y^{2}$ out of $20 \sqrt{12} a^{5} y^{7} - a y^{2} \sqrt{3 a^{5} y^{3}}$ .

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Step 1.1

Factor $a y^{2}$ out of $20 \sqrt{12} a^{5} y^{7}$ .

$a y^{2} (20 \sqrt{12} a^{4} y^{5}) - a y^{2} \sqrt{3 a^{5} y^{3}}$

Step 1.2

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

$a y^{2} (20 \sqrt{12} a^{4} y^{5}) + a y^{2} (- 1 \sqrt{3 a^{5} y^{3}})$

Step 1.3

Factor $a y^{2}$ out of $a y^{2} (20 \sqrt{12} a^{4} y^{5}) + a y^{2} (- 1 \sqrt{3 a^{5} y^{3}})$ .

$a y^{2} (20 \sqrt{12} a^{4} y^{5} - 1 \sqrt{3 a^{5} y^{3}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12}$ as $12^{\frac{1}{2}}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 \sqrt{3 a^{5} y^{3}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 a^{5} y^{3}}$ as ${(3 a^{5} y^{3})}^{\frac{1}{2}}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 {(3 a^{5} y^{3})}^{\frac{1}{2}})$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $3 a^{5} y^{3}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 ({(3 a^{5})}^{\frac{1}{2}} {(y^{3})}^{\frac{1}{2}}))$

Step 4.2

Apply the product rule to $3 a^{5}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 (3^{\frac{1}{2}} {(a^{5})}^{\frac{1}{2}} {(y^{3})}^{\frac{1}{2}}))$

Step 5

Multiply the exponents in ${(a^{5})}^{\frac{1}{2}}$ .

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Step 5.1

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

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 (3^{\frac{1}{2}} a^{5 (\frac{1}{2})} {(y^{3})}^{\frac{1}{2}}))$

Step 5.2

Combine $5$ and $\frac{1}{2}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 (3^{\frac{1}{2}} a^{\frac{5}{2}} {(y^{3})}^{\frac{1}{2}}))$

Step 6

Multiply the exponents in ${(y^{3})}^{\frac{1}{2}}$ .

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Step 6.1

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

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 (3^{\frac{1}{2}} a^{\frac{5}{2}} y^{3 (\frac{1}{2})}))$

Step 6.2

Combine $3$ and $\frac{1}{2}$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 1 (3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}}))$

Step 7

Rewrite $- 1 (3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}})$ as $- (3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}})$ .

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - (3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}}))$

Step 8

Remove parentheses.

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - (3^{\frac{1}{2}} a^{\frac{5}{2}}) y^{\frac{3}{2}})$

Step 9

Remove parentheses.

$a y^{2} (20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}})$

Step 10

Factor.

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Step 10.1

Factor $a^{\frac{5}{2}} y^{\frac{3}{2}}$ out of $20 \cdot 12^{\frac{1}{2}} a^{4} y^{5} - 3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}}$ .

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Step 10.1.1

Factor $a^{\frac{5}{2}} y^{\frac{3}{2}}$ out of $20 \cdot 12^{\frac{1}{2}} a^{4} y^{5}$ .

$a y^{2} (a^{\frac{5}{2}} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) - 3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}})$

Step 10.1.2

Factor $a^{\frac{5}{2}} y^{\frac{3}{2}}$ out of $- 3^{\frac{1}{2}} a^{\frac{5}{2}} y^{\frac{3}{2}}$ .

$a y^{2} (a^{\frac{5}{2}} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) + a^{\frac{5}{2}} y^{\frac{3}{2}} (- 3^{\frac{1}{2}}))$

Step 10.1.3

Factor $a^{\frac{5}{2}} y^{\frac{3}{2}}$ out of $a^{\frac{5}{2}} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) + a^{\frac{5}{2}} y^{\frac{3}{2}} (- 3^{\frac{1}{2}})$ .

$a y^{2} (a^{\frac{5}{2}} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}}))$

Step 10.2

Remove unnecessary parentheses.

$a y^{2} a^{\frac{5}{2}} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11

Combine exponents.

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Step 11.1

Multiply $a$ by $a^{\frac{5}{2}}$ by adding the exponents.

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Step 11.1.1

Move $a^{\frac{5}{2}}$ .

$a^{\frac{5}{2}} a y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.1.2

Multiply $a^{\frac{5}{2}}$ by $a$ .

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Step 11.1.2.1

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

$a^{\frac{5}{2}} a^{1} y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.1.2.2

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

$a^{\frac{5}{2} + 1} y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.1.3

Write $1$ as a fraction with a common denominator.

$a^{\frac{5}{2} + \frac{2}{2}} y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.1.4

Combine the numerators over the common denominator.

$a^{\frac{5 + 2}{2}} y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.1.5

Add $5$ and $2$ .

$a^{\frac{7}{2}} y^{2} y^{\frac{3}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2

Multiply $y^{2}$ by $y^{\frac{3}{2}}$ by adding the exponents.

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Step 11.2.1

Move $y^{\frac{3}{2}}$ .

$a^{\frac{7}{2}} (y^{\frac{3}{2}} y^{2}) (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.2

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

$a^{\frac{7}{2}} y^{\frac{3}{2} + 2} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a^{\frac{7}{2}} y^{\frac{3}{2} + 2 \cdot \frac{2}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.4

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

$a^{\frac{7}{2}} y^{\frac{3}{2} + \frac{2 \cdot 2}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.5

Combine the numerators over the common denominator.

$a^{\frac{7}{2}} y^{\frac{3 + 2 \cdot 2}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.6

Simplify the numerator.

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Step 11.2.6.1

Multiply $2$ by $2$ .

$a^{\frac{7}{2}} y^{\frac{3 + 4}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 11.2.6.2

Add $3$ and $4$ .

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 12

Multiply $20$ by $12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}$ .

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) - 3^{\frac{1}{2}})$

Step 13

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}}) y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 14

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 15

Multiply $20$ by $12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}$ .

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) - 3^{\frac{1}{2}})$

Step 16

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}}) y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 17

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 18

Multiply $20$ by $12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}$ .

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}}) - 3^{\frac{1}{2}})$

Step 19

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 (12^{\frac{1}{2}} a^{\frac{3}{2}}) y^{\frac{7}{2}} - 3^{\frac{1}{2}})$

Step 20

Remove parentheses.

$a^{\frac{7}{2}} y^{\frac{7}{2}} (20 \cdot 12^{\frac{1}{2}} a^{\frac{3}{2}} y^{\frac{7}{2}} - 3^{\frac{1}{2}})$