Algebra Examples

$\frac{33}{4} = n^{- \frac{1}{3}} + 8$

Step 1

Rewrite the equation as $n^{- \frac{1}{3}} + 8 = \frac{33}{4}$ .

$n^{- \frac{1}{3}} + 8 = \frac{33}{4}$

Step 2

Move all terms not containing $n$ to the right side of the equation.

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Step 2.1

Subtract $8$ from both sides of the equation.

$n^{- \frac{1}{3}} = \frac{33}{4} - 8$

Step 2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$n^{- \frac{1}{3}} = \frac{33}{4} - 8 \cdot \frac{4}{4}$

Step 2.3

Combine $- 8$ and $\frac{4}{4}$ .

$n^{- \frac{1}{3}} = \frac{33}{4} + \frac{- 8 \cdot 4}{4}$

Step 2.4

Combine the numerators over the common denominator.

$n^{- \frac{1}{3}} = \frac{33 - 8 \cdot 4}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 8$ by $4$ .

$n^{- \frac{1}{3}} = \frac{33 - 32}{4}$

Step 2.5.2

Subtract $32$ from $33$ .

$n^{- \frac{1}{3}} = \frac{1}{4}$

Step 3

Raise each side of the equation to the power of $- 3$ to eliminate the fractional exponent on the left side.

${(n^{- \frac{1}{3}})}^{- 3} = {(\frac{1}{4})}^{- 3}$

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify ${(n^{- \frac{1}{3}})}^{- 3}$ .

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Step 4.1.1.1

Multiply the exponents in ${(n^{- \frac{1}{3}})}^{- 3}$ .

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Step 4.1.1.1.1

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

$n^{- \frac{1}{3} \cdot - 3} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$n^{\frac{- 1}{3} \cdot - 3} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.1.2.2

Factor $3$ out of $- 3$ .

$n^{\frac{- 1}{3} \cdot (3 (- 1))} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.1.2.3

Cancel the common factor.

$n^{\frac{- 1}{3} \cdot (3 \cdot - 1)} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.1.2.4

Rewrite the expression.

$n^{- 1 \cdot - 1} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.1.3

Multiply $- 1$ by $- 1$ .

$n^{1} = {(\frac{1}{4})}^{- 3}$

Step 4.1.1.2

Simplify.

$n = {(\frac{1}{4})}^{- 3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(\frac{1}{4})}^{- 3}$ .

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Step 4.2.1.1

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

$n = 4^{3}$

Step 4.2.1.2

Raise $4$ to the power of $3$ .

$n = 64$