Finite Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $a b^{3}$ as $b^{2} (a b)$ .

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Step 1.1.1

Factor out $b^{2}$ .

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

Step 1.1.2

Reorder $a$ and $b^{2}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Cancel the common factor of $b$ .

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Step 1.3.1

Move the leading negative in $- \frac{1}{b}$ into the numerator.

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

Step 1.3.2

Factor $b$ out of $b \sqrt{a b}$ .

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

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 $a \sqrt{a}$ .

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