Basic Math Examples

$\frac{1}{2} \cdot \sqrt{3 - \frac{3}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1

Simplify each term.

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Step 1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{2} \cdot \sqrt{3 \cdot \frac{4}{4} - \frac{3}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.2

Combine $3$ and $\frac{4}{4}$ .

$\frac{1}{2} \cdot \sqrt{\frac{3 \cdot 4}{4} - \frac{3}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \sqrt{\frac{3 \cdot 4 - 3}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Multiply $3$ by $4$ .

$\frac{1}{2} \cdot \sqrt{\frac{12 - 3}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.4.2

Subtract $3$ from $12$ .

$\frac{1}{2} \cdot \sqrt{\frac{9}{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.5

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$\frac{1}{2} \cdot \frac{\sqrt{9}}{\sqrt{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

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

$\frac{1}{2} \cdot \frac{\sqrt{3^{2}}}{\sqrt{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.6.2

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

$\frac{1}{2} \cdot \frac{3}{\sqrt{4}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.7

Simplify the denominator.

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Step 1.7.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{2^{2}}} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.7.2

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

$\frac{1}{2} \cdot \frac{3}{2} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.8

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

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Step 1.8.1

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

$\frac{3}{2 \cdot 2} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.8.2

Multiply $2$ by $2$ .

$\frac{3}{4} + {(2 - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.9

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{4} + {(2 \cdot \frac{3}{3} - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.10

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

$\frac{3}{4} + {(\frac{2 \cdot 3}{3} - \frac{2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.11

Combine the numerators over the common denominator.

$\frac{3}{4} + {(\frac{2 \cdot 3 - 2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.12

Simplify the numerator.

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Step 1.12.1

Multiply $2$ by $3$ .

$\frac{3}{4} + {(\frac{6 - 2}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.12.2

Subtract $2$ from $6$ .

$\frac{3}{4} + {(\frac{4}{3})}^{- 2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.13

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{3}{4} + {(\frac{3}{4})}^{2} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.14

Apply the product rule to $\frac{3}{4}$ .

$\frac{3}{4} + \frac{3^{2}}{4^{2}} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.15

Raise $3$ to the power of $2$ .

$\frac{3}{4} + \frac{9}{4^{2}} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.16

Raise $4$ to the power of $2$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{\frac{1}{4}}{{(\frac{2}{3})}^{0}}}$

Step 1.17

Apply the product rule to $\frac{2}{3}$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{\frac{1}{4}}{\frac{2^{0}}{3^{0}}}}$

Step 1.18

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{1}{4} \cdot \frac{3^{0}}{2^{0}}}$

Step 1.19

Combine.

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{1 \cdot 3^{0}}{4 \cdot 2^{0}}}$

Step 1.20

Multiply $3^{0}$ by $1$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{3^{0}}{4 \cdot 2^{0}}}$

Step 1.21

Simplify the denominator.

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Step 1.21.1

Rewrite $4$ as $2^{2}$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{3^{0}}{2^{2} \cdot 2^{0}}}$

Step 1.21.2

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

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{3^{0}}{2^{2 + 0}}}$

Step 1.21.3

Add $2$ and $0$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{3^{0}}{2^{2}}}$

Step 1.22

Anything raised to $0$ is $1$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{1}{2^{2}}}$

Step 1.23

Raise $2$ to the power of $2$ .

$\frac{3}{4} + \frac{9}{16} - \sqrt{\frac{1}{4}}$

Step 1.24

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{3}{4} + \frac{9}{16} - \frac{\sqrt{1}}{\sqrt{4}}$

Step 1.25

Any root of $1$ is $1$ .

$\frac{3}{4} + \frac{9}{16} - \frac{1}{\sqrt{4}}$

Step 1.26

Simplify the denominator.

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Step 1.26.1

Rewrite $4$ as $2^{2}$ .

$\frac{3}{4} + \frac{9}{16} - \frac{1}{\sqrt{2^{2}}}$

Step 1.26.2

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

$\frac{3}{4} + \frac{9}{16} - \frac{1}{2}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{3}{4}$ by $\frac{4}{4}$ .

$\frac{3}{4} \cdot \frac{4}{4} + \frac{9}{16} - \frac{1}{2}$

Step 2.2

Multiply $\frac{3}{4}$ by $\frac{4}{4}$ .

$\frac{3 \cdot 4}{4 \cdot 4} + \frac{9}{16} - \frac{1}{2}$

Step 2.3

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$\frac{3 \cdot 4}{4 \cdot 4} + \frac{9}{16} - (\frac{1}{2} \cdot \frac{8}{8})$

Step 2.4

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$\frac{3 \cdot 4}{4 \cdot 4} + \frac{9}{16} - \frac{8}{2 \cdot 8}$

Step 2.5

Multiply $4$ by $4$ .

$\frac{3 \cdot 4}{16} + \frac{9}{16} - \frac{8}{2 \cdot 8}$

Step 2.6

Multiply $2$ by $8$ .

$\frac{3 \cdot 4}{16} + \frac{9}{16} - \frac{8}{16}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 \cdot 4 + 9 - 8}{16}$

Step 4

Simplify the expression.

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Step 4.1

Multiply $3$ by $4$ .

$\frac{12 + 9 - 8}{16}$

Step 4.2

Add $12$ and $9$ .

$\frac{21 - 8}{16}$

Step 4.3

Subtract $8$ from $21$ .

$\frac{13}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{16}$

Decimal Form:

$0.8125$