Pre-Algebra Examples

$\frac{\sqrt{{(a + b)}^{5} r^{2} y^{3}}}{\sqrt{0.25 (a + b) r^{- 6} y^{5}}}$

Step 1

Combine $\sqrt{{(a + b)}^{5} r^{2} y^{3}}$ and $\sqrt{0.25 (a + b) r^{- 6} y^{5}}$ into a single radical.

$\sqrt{\frac{{(a + b)}^{5} r^{2} y^{3}}{0.25 (a + b) r^{- 6} y^{5}}}$

Step 2

Reduce the expression $\frac{{(a + b)}^{5} r^{2} y^{3}}{0.25 (a + b) r^{- 6} y^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $a + b$ out of ${(a + b)}^{5} r^{2} y^{3}$ .

$\sqrt{\frac{(a + b) ({(a + b)}^{4} r^{2} y^{3})}{0.25 (a + b) r^{- 6} y^{5}}}$

Step 2.2

Factor $a + b$ out of $0.25 (a + b) r^{- 6} y^{5}$ .

$\sqrt{\frac{(a + b) ({(a + b)}^{4} r^{2} y^{3})}{(a + b) (0.25 r^{- 6} y^{5})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{(a + b) ({(a + b)}^{4} r^{2} y^{3})}{(a + b) (0.25 r^{- 6} y^{5})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{{(a + b)}^{4} r^{2} y^{3}}{0.25 r^{- 6} y^{5}}}$

Step 3

Move $r^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt{\frac{{(a + b)}^{4} r^{2} y^{3} r^{6}}{0.25 y^{5}}}$

Step 4

Multiply $r^{2}$ by $r^{6}$ by adding the exponents.

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Step 4.1

Move $r^{6}$ .

$\sqrt{\frac{{(a + b)}^{4} (r^{6} r^{2}) y^{3}}{0.25 y^{5}}}$

Step 4.2

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

$\sqrt{\frac{{(a + b)}^{4} r^{6 + 2} y^{3}}{0.25 y^{5}}}$

Step 4.3

Add $6$ and $2$ .

$\sqrt{\frac{{(a + b)}^{4} r^{8} y^{3}}{0.25 y^{5}}}$

Step 5

Cancel the common factor of $y^{3}$ and $y^{5}$ .

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Step 5.1

Factor $y^{3}$ out of ${(a + b)}^{4} r^{8} y^{3}$ .

$\sqrt{\frac{y^{3} ({(a + b)}^{4} r^{8})}{0.25 y^{5}}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $y^{3}$ out of $0.25 y^{5}$ .

$\sqrt{\frac{y^{3} ({(a + b)}^{4} r^{8})}{y^{3} (0.25 y^{2})}}$

Step 5.2.2

Cancel the common factor.

$\sqrt{\frac{y^{3} ({(a + b)}^{4} r^{8})}{y^{3} (0.25 y^{2})}}$

Step 5.2.3

Rewrite the expression.

$\sqrt{\frac{{(a + b)}^{4} r^{8}}{0.25 y^{2}}}$

Step 6

Factor $1$ out of ${(a + b)}^{4} r^{8}$ .

$\sqrt{\frac{1 ({(a + b)}^{4} r^{8})}{0.25 y^{2}}}$

Step 7

Factor $0.25$ out of $0.25 y^{2}$ .

$\sqrt{\frac{1 ({(a + b)}^{4} r^{8})}{0.25 (y^{2})}}$

Step 8

Separate fractions.

$\sqrt{\frac{1}{0.25} \cdot \frac{{(a + b)}^{4} r^{8}}{y^{2}}}$

Step 9

Divide $1$ by $0.25$ .

$\sqrt{4 \frac{{(a + b)}^{4} r^{8}}{y^{2}}}$

Step 10

Combine $4$ and $\frac{{(a + b)}^{4} r^{8}}{y^{2}}$ .

$\sqrt{\frac{4 ({(a + b)}^{4} r^{8})}{y^{2}}}$

Step 11

Rewrite $4 {(a + b)}^{4} r^{8}$ as ${(2 {(a + b)}^{2} r^{4})}^{2}$ .

$\sqrt{\frac{{(2 {(a + b)}^{2} r^{4})}^{2}}{y^{2}}}$

Step 12

Rewrite $\frac{{(2 {(a + b)}^{2} r^{4})}^{2}}{y^{2}}$ as ${(\frac{2 {(a + b)}^{2} r^{4}}{y})}^{2}$ .

$\sqrt{{(\frac{2 {(a + b)}^{2} r^{4}}{y})}^{2}}$

Step 13

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 {(a + b)}^{2} r^{4}}{y}$

Step 14

Reorder factors in $\frac{2 {(a + b)}^{2} r^{4}}{y}$ .

$\frac{2 r^{4} {(a + b)}^{2}}{y}$

Step 15

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

$\frac{2 r^{4} ((a + b) (a + b))}{y}$

Step 16

Expand $(a + b) (a + b)$ using the FOIL Method.

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Step 16.1

Apply the distributive property.

$\frac{2 r^{4} (a (a + b) + b (a + b))}{y}$

Step 16.2

Apply the distributive property.

$\frac{2 r^{4} (a \cdot a + a b + b (a + b))}{y}$

Step 16.3

Apply the distributive property.

$\frac{2 r^{4} (a \cdot a + a b + b a + b \cdot b)}{y}$

Step 17

Simplify and combine like terms.

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Step 17.1

Simplify each term.

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Step 17.1.1

Multiply $a$ by $a$ .

$\frac{2 r^{4} (a^{2} + a b + b a + b \cdot b)}{y}$

Step 17.1.2

Multiply $b$ by $b$ .

$\frac{2 r^{4} (a^{2} + a b + b a + b^{2})}{y}$

Step 17.2

Add $a b$ and $b a$ .

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Step 17.2.1

Reorder $b$ and $a$ .

$\frac{2 r^{4} (a^{2} + a b + a b + b^{2})}{y}$

Step 17.2.2

Add $a b$ and $a b$ .

$\frac{2 r^{4} (a^{2} + 2 a b + b^{2})}{y}$

Step 18

Apply the distributive property.

$\frac{2 r^{4} a^{2} + 2 r^{4} (2 a b) + 2 r^{4} b^{2}}{y}$

Step 19

Multiply $2$ by $2$ .

$\frac{2 r^{4} a^{2} + 4 r^{4} a b + 2 r^{4} b^{2}}{y}$

Step 20

Split the fraction $\frac{2 r^{4} a^{2} + 4 r^{4} a b + 2 r^{4} b^{2}}{y}$ into two fractions.

$\frac{2 r^{4} a^{2} + 4 r^{4} a b}{y} + \frac{2 r^{4} b^{2}}{y}$

Step 21

Split the fraction $\frac{2 r^{4} a^{2} + 4 r^{4} a b}{y}$ into two fractions.

$\frac{2 r^{4} a^{2}}{y} + \frac{4 r^{4} a b}{y} + \frac{2 r^{4} b^{2}}{y}$