Basic Math Examples

$(3^{- 1} + \frac{1}{2} \cdot (\frac{1}{2} - \frac{1}{4}) + 5) - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} + \frac{1}{2} \cdot (\frac{1}{2} - \frac{1}{4}) + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3} + \frac{1}{2} \cdot (\frac{1}{2} \cdot \frac{2}{2} - \frac{1}{4}) + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

$\frac{1}{3} + \frac{1}{2} \cdot (\frac{2}{2 \cdot 2} - \frac{1}{4}) + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.3.2

Multiply $2$ by $2$ .

$\frac{1}{3} + \frac{1}{2} \cdot (\frac{2}{4} - \frac{1}{4}) + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{1}{3} + \frac{1}{2} \cdot \frac{2 - 1}{4} + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.5

Subtract $1$ from $2$ .

$\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{4} + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.6

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

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Step 1.6.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{3} + \frac{1}{2 \cdot 4} + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.6.2

Multiply $2$ by $4$ .

$\frac{1}{3} + \frac{1}{8} + 5 - 34 \div \sqrt{\frac{9}{25} + 1}$

Step 1.7

Rewrite the division as a fraction.

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\sqrt{\frac{9}{25} + 1}}$

Step 1.8

Simplify the denominator.

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Step 1.8.1

Write $1$ as a fraction with a common denominator.

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\sqrt{\frac{9}{25} + \frac{25}{25}}}$

Step 1.8.2

Combine the numerators over the common denominator.

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\sqrt{\frac{9 + 25}{25}}}$

Step 1.8.3

Add $9$ and $25$ .

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\sqrt{\frac{34}{25}}}$

Step 1.8.4

Rewrite $\sqrt{\frac{34}{25}}$ as $\frac{\sqrt{34}}{\sqrt{25}}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\frac{\sqrt{34}}{\sqrt{25}}}$

Step 1.8.5

Simplify the denominator.

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Step 1.8.5.1

Rewrite $25$ as $5^{2}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\frac{\sqrt{34}}{\sqrt{5^{2}}}}$

Step 1.8.5.2

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

$\frac{1}{3} + \frac{1}{8} + 5 - \frac{34}{\frac{\sqrt{34}}{5}}$

Step 1.9

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5}{\sqrt{34}})$

Step 1.10

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 (\frac{5}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}))$

Step 1.11

Combine and simplify the denominator.

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Step 1.11.1

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}})$

Step 1.11.2

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{{\sqrt{34}}^{1} \sqrt{34}})$

Step 1.11.3

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{{\sqrt{34}}^{1} {\sqrt{34}}^{1}})$

Step 1.11.4

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

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{{\sqrt{34}}^{1 + 1}})$

Step 1.11.5

Add $1$ and $1$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{{\sqrt{34}}^{2}})$

Step 1.11.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 1.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}})$

Step 1.11.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}})$

Step 1.11.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34^{\frac{2}{2}}})$

Step 1.11.6.4

Cancel the common factor of $2$ .

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Step 1.11.6.4.1

Cancel the common factor.

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34^{\frac{2}{2}}})$

Step 1.11.6.4.2

Rewrite the expression.

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34^{1}})$

Step 1.11.6.5

Evaluate the exponent.

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34})$

Step 1.12

Cancel the common factor of $34$ .

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Step 1.12.1

Cancel the common factor.

$\frac{1}{3} + \frac{1}{8} + 5 - (34 \frac{5 \sqrt{34}}{34})$

Step 1.12.2

Rewrite the expression.

$\frac{1}{3} + \frac{1}{8} + 5 - (5 \sqrt{34})$

Step 1.13

Multiply $5$ by $- 1$ .

$\frac{1}{3} + \frac{1}{8} + 5 - 5 \sqrt{34}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{1}{3}$ by $\frac{8}{8}$ .

$\frac{1}{3} \cdot \frac{8}{8} + \frac{1}{8} + 5 - 5 \sqrt{34}$

Step 2.2

Multiply $\frac{1}{3}$ by $\frac{8}{8}$ .

$\frac{8}{3 \cdot 8} + \frac{1}{8} + 5 - 5 \sqrt{34}$

Step 2.3

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$\frac{8}{3 \cdot 8} + \frac{1}{8} \cdot \frac{3}{3} + 5 - 5 \sqrt{34}$

Step 2.4

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + 5 - 5 \sqrt{34}$

Step 2.5

Write $5$ as a fraction with denominator $1$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5}{1} - 5 \sqrt{34}$

Step 2.6

Multiply $\frac{5}{1}$ by $\frac{24}{24}$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5}{1} \cdot \frac{24}{24} - 5 \sqrt{34}$

Step 2.7

Multiply $\frac{5}{1}$ by $\frac{24}{24}$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5 \cdot 24}{24} - 5 \sqrt{34}$

Step 2.8

Write $- 5 \sqrt{34}$ as a fraction with denominator $1$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34}}{1}$

Step 2.9

Multiply $\frac{- 5 \sqrt{34}}{1}$ by $\frac{24}{24}$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34}}{1} \cdot \frac{24}{24}$

Step 2.10

Multiply $\frac{- 5 \sqrt{34}}{1}$ by $\frac{24}{24}$ .

$\frac{8}{3 \cdot 8} + \frac{3}{8 \cdot 3} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34} \cdot 24}{24}$

Step 2.11

Multiply $3$ by $8$ .

$\frac{8}{24} + \frac{3}{8 \cdot 3} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34} \cdot 24}{24}$

Step 2.12

Reorder the factors of $8 \cdot 3$ .

$\frac{8}{24} + \frac{3}{3 \cdot 8} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34} \cdot 24}{24}$

Step 2.13

Multiply $3$ by $8$ .

$\frac{8}{24} + \frac{3}{24} + \frac{5 \cdot 24}{24} + \frac{- 5 \sqrt{34} \cdot 24}{24}$

Step 3

Combine the numerators over the common denominator.

$\frac{8 + 3 + 5 \cdot 24 - 5 \sqrt{34} \cdot 24}{24}$

Step 4

Simplify each term.

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Step 4.1

Multiply $5$ by $24$ .

$\frac{8 + 3 + 120 - 5 \sqrt{34} \cdot 24}{24}$

Step 4.2

Multiply $24$ by $- 5$ .

$\frac{8 + 3 + 120 - 120 \sqrt{34}}{24}$

Step 5

Simplify by adding numbers.

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Step 5.1

Add $8$ and $3$ .

$\frac{11 + 120 - 120 \sqrt{34}}{24}$

Step 5.2

Add $11$ and $120$ .

$\frac{131 - 120 \sqrt{34}}{24}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{131 - 120 \sqrt{34}}{24}$

Decimal Form:

$- 23.69642614 \dots$