Evaluate (3^-1+1/2*(1/2-1/4)+5)-34÷( square root of 9/25+1)

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

Step 1

Simplify each term.

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

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

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

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

Multiply $2$ by $4$ .

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

Rewrite the division as a fraction.

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

Simplify the denominator.

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

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

Combine the numerators over the common denominator.

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

Add $9$ and $25$ .

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

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

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

Simplify the denominator.

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $34$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $5$ by $- 1$ .

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

Step 2

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $8$ .

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

Reorder the factors of $8 \cdot 3$ .

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

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.

Multiply $5$ by $24$ .

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

Multiply $24$ by $- 5$ .

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

Step 5

Simplify by adding numbers.

Add $8$ and $3$ .

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

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:

$- 23.69642614 \dots$

Evaluate (3^-1+1/2*(1/2-1/4)+5)-34÷( square root of 9/25+1)

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