Basic Math Examples

$\frac{\frac{2}{3} + \frac{2}{5} - \frac{2}{9}}{\frac{4}{3} + \frac{4}{5} - \frac{4}{9}} = t \sqrt{9}$

Step 1

Rewrite the equation as $t \sqrt{9} = \frac{\frac{2}{3} + \frac{2}{5} - \frac{2}{9}}{\frac{4}{3} + \frac{4}{5} - \frac{4}{9}}$ .

$t \sqrt{9} = \frac{\frac{2}{3} + \frac{2}{5} - \frac{2}{9}}{\frac{4}{3} + \frac{4}{5} - \frac{4}{9}}$

Step 2

Simplify.

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Step 2.1

Multiply the numerator and denominator of the fraction by $45$ .

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Step 2.1.1

Multiply $\frac{\frac{2}{3} + \frac{2}{5} - \frac{2}{9}}{\frac{4}{3} + \frac{4}{5} - \frac{4}{9}}$ by $\frac{45}{45}$ .

$t \sqrt{9} = \frac{45}{45} \cdot \frac{\frac{2}{3} + \frac{2}{5} - \frac{2}{9}}{\frac{4}{3} + \frac{4}{5} - \frac{4}{9}}$

Step 2.1.2

Combine.

$t \sqrt{9} = \frac{45 (\frac{2}{3} + \frac{2}{5} - \frac{2}{9})}{45 (\frac{4}{3} + \frac{4}{5} - \frac{4}{9})}$

Step 2.2

Apply the distributive property.

$t \sqrt{9} = \frac{45 (\frac{2}{3}) + 45 (\frac{2}{5}) + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $3$ .

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Step 2.3.1.1

Factor $3$ out of $45$ .

$t \sqrt{9} = \frac{3 (15) \frac{2}{3} + 45 (\frac{2}{5}) + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.1.2

Cancel the common factor.

$t \sqrt{9} = \frac{3 \cdot 15 \frac{2}{3} + 45 (\frac{2}{5}) + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.1.3

Rewrite the expression.

$t \sqrt{9} = \frac{15 \cdot 2 + 45 (\frac{2}{5}) + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.2

Multiply $15$ by $2$ .

$t \sqrt{9} = \frac{30 + 45 (\frac{2}{5}) + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.3

Cancel the common factor of $5$ .

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Step 2.3.3.1

Factor $5$ out of $45$ .

$t \sqrt{9} = \frac{30 + 5 (9) \frac{2}{5} + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.3.2

Cancel the common factor.

$t \sqrt{9} = \frac{30 + 5 \cdot 9 \frac{2}{5} + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.3.3

Rewrite the expression.

$t \sqrt{9} = \frac{30 + 9 \cdot 2 + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.4

Multiply $9$ by $2$ .

$t \sqrt{9} = \frac{30 + 18 + 45 (- \frac{2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.5

Cancel the common factor of $9$ .

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Step 2.3.5.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

$t \sqrt{9} = \frac{30 + 18 + 45 (\frac{- 2}{9})}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.5.2

Factor $9$ out of $45$ .

$t \sqrt{9} = \frac{30 + 18 + 9 (5) \frac{- 2}{9}}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.5.3

Cancel the common factor.

$t \sqrt{9} = \frac{30 + 18 + 9 \cdot 5 \frac{- 2}{9}}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.5.4

Rewrite the expression.

$t \sqrt{9} = \frac{30 + 18 + 5 \cdot - 2}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.6

Multiply $5$ by $- 2$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{45 (\frac{4}{3}) + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.7

Cancel the common factor of $3$ .

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Step 2.3.7.1

Factor $3$ out of $45$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{3 (15) \frac{4}{3} + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.7.2

Cancel the common factor.

$t \sqrt{9} = \frac{30 + 18 - 10}{3 \cdot 15 \frac{4}{3} + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.7.3

Rewrite the expression.

$t \sqrt{9} = \frac{30 + 18 - 10}{15 \cdot 4 + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.8

Multiply $15$ by $4$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 45 (\frac{4}{5}) + 45 (- \frac{4}{9})}$

Step 2.3.9

Cancel the common factor of $5$ .

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Step 2.3.9.1

Factor $5$ out of $45$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 5 (9) \frac{4}{5} + 45 (- \frac{4}{9})}$

Step 2.3.9.2

Cancel the common factor.

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 5 \cdot 9 \frac{4}{5} + 45 (- \frac{4}{9})}$

Step 2.3.9.3

Rewrite the expression.

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 9 \cdot 4 + 45 (- \frac{4}{9})}$

Step 2.3.10

Multiply $9$ by $4$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 + 45 (- \frac{4}{9})}$

Step 2.3.11

Cancel the common factor of $9$ .

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Step 2.3.11.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 + 45 (\frac{- 4}{9})}$

Step 2.3.11.2

Factor $9$ out of $45$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 + 9 (5) \frac{- 4}{9}}$

Step 2.3.11.3

Cancel the common factor.

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 + 9 \cdot 5 \frac{- 4}{9}}$

Step 2.3.11.4

Rewrite the expression.

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 + 5 \cdot - 4}$

Step 2.3.12

Multiply $5$ by $- 4$ .

$t \sqrt{9} = \frac{30 + 18 - 10}{60 + 36 - 20}$

Step 2.4

Cancel the common factor of $30 + 18 - 10$ and $60 + 36 - 20$ .

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Step 2.4.1

Factor $2$ out of $30$ .

$t \sqrt{9} = \frac{2 (15) + 18 - 10}{60 + 36 - 20}$

Step 2.4.2

Factor $2$ out of $18$ .

$t \sqrt{9} = \frac{2 \cdot 15 + 2 \cdot 9 - 10}{60 + 36 - 20}$

Step 2.4.3

Factor $2$ out of $2 \cdot 15 + 2 \cdot 9$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9) - 10}{60 + 36 - 20}$

Step 2.4.4

Factor $2$ out of $- 10$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9) + 2 (- 5)}{60 + 36 - 20}$

Step 2.4.5

Factor $2$ out of $2 \cdot (15 + 9) + 2 (- 5)$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{60 + 36 - 20}$

Step 2.4.6

Cancel the common factors.

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Step 2.4.6.1

Factor $2$ out of $60$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 (30) + 36 - 20}$

Step 2.4.6.2

Factor $2$ out of $36$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 \cdot 30 + 2 \cdot 18 - 20}$

Step 2.4.6.3

Factor $2$ out of $2 \cdot 30 + 2 \cdot 18$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 \cdot (30 + 18) - 20}$

Step 2.4.6.4

Factor $2$ out of $- 20$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 \cdot (30 + 18) + 2 (- 10)}$

Step 2.4.6.5

Factor $2$ out of $2 \cdot (30 + 18) + 2 (- 10)$ .

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 \cdot (30 + 18 - 10)}$

Step 2.4.6.6

Cancel the common factor.

$t \sqrt{9} = \frac{2 \cdot (15 + 9 - 5)}{2 \cdot (30 + 18 - 10)}$

Step 2.4.6.7

Rewrite the expression.

$t \sqrt{9} = \frac{15 + 9 - 5}{30 + 18 - 10}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Add $15$ and $9$ .

$t \sqrt{9} = \frac{24 - 5}{30 + 18 - 10}$

Step 2.5.2

Subtract $5$ from $24$ .

$t \sqrt{9} = \frac{19}{30 + 18 - 10}$

Step 2.6

Simplify the denominator.

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Step 2.6.1

Add $30$ and $18$ .

$t \sqrt{9} = \frac{19}{48 - 10}$

Step 2.6.2

Subtract $10$ from $48$ .

$t \sqrt{9} = \frac{19}{38}$

Step 2.7

Cancel the common factor of $19$ and $38$ .

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Step 2.7.1

Factor $19$ out of $19$ .

$t \sqrt{9} = \frac{19 (1)}{38}$

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $19$ out of $38$ .

$t \sqrt{9} = \frac{19 \cdot 1}{19 \cdot 2}$

Step 2.7.2.2

Cancel the common factor.

$t \sqrt{9} = \frac{19 \cdot 1}{19 \cdot 2}$

Step 2.7.2.3

Rewrite the expression.

$t \sqrt{9} = \frac{1}{2}$

Step 3

Divide each term in $t \sqrt{9} = \frac{1}{2}$ by $\sqrt{9}$ and simplify.

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Step 3.1

Divide each term in $t \sqrt{9} = \frac{1}{2}$ by $\sqrt{9}$ .

$\frac{t \sqrt{9}}{\sqrt{9}} = \frac{\frac{1}{2}}{\sqrt{9}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sqrt{9}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{t \sqrt{9}}{\sqrt{9}} = \frac{\frac{1}{2}}{\sqrt{9}}$

Step 3.2.1.2

Divide $t$ by $1$ .

$t = \frac{\frac{1}{2}}{\sqrt{9}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{1}{2} \cdot \frac{1}{\sqrt{9}}$

Step 3.3.2

Simplify the denominator.

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Step 3.3.2.1

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

$t = \frac{1}{2} \cdot \frac{1}{\sqrt{3^{2}}}$

Step 3.3.2.2

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

$t = \frac{1}{2} \cdot \frac{1}{3}$

Step 3.3.3

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 3.3.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$t = \frac{1}{2 \cdot 3}$

Step 3.3.3.2

Multiply $2$ by $3$ .

$t = \frac{1}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{1}{6}$

Decimal Form:

$t = 0.1\overline{6}$