Finite Math Examples

\sqrt{a b} - \frac{1}{b} \cdot \sqrt{a b^{3}} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot \sqrt{a b^{3}} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1

Simplify each term.

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Step 1.1

Rewrite

a b^{3}

$a b^{3}$ as

b^{2} (a b)

$b^{2} (a b)$ .

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Step 1.1.1

Factor out

b^{2}

$b^{2}$ .

\sqrt{a b} - \frac{1}{b} \cdot \sqrt{a (b^{2} b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot \sqrt{a (b^{2} b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.1.2

Reorder

a

$a$ and

b^{2}

$b^{2}$ .

\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} a b} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} a b} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.1.3

Add parentheses.

\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} (a b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} (a b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} (a b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot \sqrt{b^{2} (a b)} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.2

Pull terms out from under the radical.

\sqrt{a b} - \frac{1}{b} \cdot (b \sqrt{a b}) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} - \frac{1}{b} \cdot (b \sqrt{a b}) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.3

Cancel the common factor of

b

$b$ .

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Step 1.3.1

Move the leading negative in

- \frac{1}{b}

$- \frac{1}{b}$ into the numerator.

\sqrt{a b} + \frac{- 1}{b} \cdot (b \sqrt{a b}) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} + \frac{- 1}{b} \cdot (b \sqrt{a b}) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.3.2

Factor

b

$b$ out of

b \sqrt{a b}

$b \sqrt{a b}$ .

\sqrt{a b} + \frac{- 1}{b} \cdot (b (\sqrt{a b})) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})

$\sqrt{a b} + \frac{- 1}{b} \cdot (b (\sqrt{a b})) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.3.3

Cancel the common factor.

$\sqrt{a b} + \frac{- 1}{b} \cdot (b \sqrt{a b}) - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.3.4

Rewrite the expression.

$\sqrt{a b} - 1 \cdot \sqrt{a b} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.4

Rewrite

$- 1 \sqrt{a b}$ as

$- \sqrt{a b}$ .

$\sqrt{a b} - \sqrt{a b} - a \sqrt{\frac{49 a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.5

Rewrite

$\frac{49 a}{9}$ as

${(\frac{7}{3})}^{2} a$ .

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Step 1.5.1

Factor the perfect power

$7^{2}$ out of

$49 a$ .

$\sqrt{a b} - \sqrt{a b} - a \sqrt{\frac{7^{2} a}{9}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.5.2

Factor the perfect power

$3^{2}$ out of

$9$ .

$\sqrt{a b} - \sqrt{a b} - a \sqrt{\frac{7^{2} a}{3^{2} \cdot 1}} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.5.3

Rearrange the fraction

$\frac{7^{2} a}{3^{2} \cdot 1}$ .

$\sqrt{a b} - \sqrt{a b} - a \sqrt{{(\frac{7}{3})}^{2} a} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.6

Pull terms out from under the radical.

$\sqrt{a b} - \sqrt{a b} - a (\frac{7}{3} \sqrt{a}) + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.7

Combine

$\frac{7}{3}$ and

$\sqrt{a}$ .

$\sqrt{a b} - \sqrt{a b} - a \frac{7 \sqrt{a}}{3} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.8

Combine

$\frac{7 \sqrt{a}}{3}$ and

$a$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + \sqrt{81 a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.9

Rewrite

$81 a^{3}$ as

${(9 a)}^{2} a$ .

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Step 1.9.1

Rewrite

$81$ as

$9^{2}$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + \sqrt{9^{2} a^{3}} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.9.2

Factor out

$a^{2}$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + \sqrt{9^{2} (a^{2} a)} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.9.3

Rewrite

$9^{2} a^{2}$ as

${(9 a)}^{2}$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + \sqrt{{(9 a)}^{2} a} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.10

Pull terms out from under the radical.

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{10}{3} \cdot (a \sqrt{a})$

Step 1.11

Multiply

$- \frac{10}{3} (a \sqrt{a})$ .

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Step 1.11.1

Combine

$a$ and

$\frac{10}{3}$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{a \cdot 10}{3} \sqrt{a}$

Step 1.11.2

Combine

$\sqrt{a}$ and

$\frac{a \cdot 10}{3}$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{\sqrt{a} (a \cdot 10)}{3}$

Step 1.12

Remove unnecessary parentheses.

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{\sqrt{a} a \cdot 10}{3}$

Step 1.13

Move

$10$ to the left of

$\sqrt{a} a$ .

$\sqrt{a b} - \sqrt{a b} - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{10 \sqrt{a} a}{3}$

Step 2

Simplify terms.

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Step 2.1

Subtract

$\sqrt{a b}$ from

$\sqrt{a b}$ .

$0 - \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{10 \sqrt{a} a}{3}$

Step 2.2

Subtract

$\frac{7 \sqrt{a} a}{3}$ from

$0$ .

$- \frac{7 \sqrt{a} a}{3} + 9 a \sqrt{a} - \frac{10 \sqrt{a} a}{3}$

Step 2.3

Combine the numerators over the common denominator.

$9 a \sqrt{a} + \frac{- 7 \sqrt{a} a - 10 \sqrt{a} a}{3}$

Step 2.4

Subtract

$10 \sqrt{a} a$ from

$- 7 \sqrt{a} a$ .

$9 a \sqrt{a} + \frac{- 17 \sqrt{a} a}{3}$

Step 2.5

Move the negative in front of the fraction.

$9 a \sqrt{a} - \frac{17 \sqrt{a} a}{3}$

Step 3

To write

$9 a \sqrt{a}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$9 a \sqrt{a} \cdot \frac{3}{3} - \frac{17 \sqrt{a} a}{3}$

Step 4

Simplify terms.

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Step 4.1

Combine

$9 a \sqrt{a}$ and

$\frac{3}{3}$ .

$\frac{9 a \sqrt{a} \cdot 3}{3} - \frac{17 \sqrt{a} a}{3}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{9 a \sqrt{a} \cdot 3 - 17 \sqrt{a} a}{3}$

Step 5

Simplify the numerator.

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Step 5.1

Factor

$a \sqrt{a}$ out of

$9 a \sqrt{a} \cdot 3 - 17 \sqrt{a} a$ .

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Step 5.1.1

Factor

$a \sqrt{a}$ out of

$9 a \sqrt{a} \cdot 3$ .

$\frac{a \sqrt{a} (9 \cdot 3) - 17 \sqrt{a} a}{3}$

Step 5.1.2

Factor

$a \sqrt{a}$ out of

$- 17 \sqrt{a} a$ .

$\frac{a \sqrt{a} (9 \cdot 3) + a \sqrt{a} (- 17)}{3}$

Step 5.1.3

Factor

$a \sqrt{a}$ out of

$a \sqrt{a} (9 \cdot 3) + a \sqrt{a} (- 17)$ .

$\frac{a \sqrt{a} (9 \cdot 3 - 17)}{3}$

Step 5.2

Multiply

$9$ by

$3$ .

$\frac{a \sqrt{a} (27 - 17)}{3}$

Step 5.3

Subtract

$17$ from

$27$ .

$\frac{a \sqrt{a} \cdot 10}{3}$

Step 6

Move

$10$ to the left of

$a \sqrt{a}$ .

$\frac{10 a \sqrt{a}}{3}$