Precalculus Examples

$5 = 3 + 4 a^{- \frac{1}{6}}$

Step 1

Rewrite the equation as $3 + 4 a^{- \frac{1}{6}} = 5$ .

$3 + 4 a^{- \frac{1}{6}} = 5$

Step 2

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

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Step 2.1

Subtract $3$ from both sides of the equation.

$4 a^{- \frac{1}{6}} = 5 - 3$

Step 2.2

Subtract $3$ from $5$ .

$4 a^{- \frac{1}{6}} = 2$

Step 3

Raise each side of the equation to the power of $- 6$ to eliminate the fractional exponent on the left side.

${(4 a^{- \frac{1}{6}})}^{- 6} = 2^{- 6}$

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 ${(4 a^{- \frac{1}{6}})}^{- 6}$ .

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Step 4.1.1.1

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

${(4 \frac{1}{a^{\frac{1}{6}}})}^{- 6} = 2^{- 6}$

Step 4.1.1.2

Combine $4$ and $\frac{1}{a^{\frac{1}{6}}}$ .

${(\frac{4}{a^{\frac{1}{6}}})}^{- 6} = 2^{- 6}$

Step 4.1.1.3

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

${(\frac{a^{\frac{1}{6}}}{4})}^{6} = 2^{- 6}$

Step 4.1.1.4

Apply the product rule to $\frac{a^{\frac{1}{6}}}{4}$ .

$\frac{{(a^{\frac{1}{6}})}^{6}}{4^{6}} = 2^{- 6}$

Step 4.1.1.5

Simplify the numerator.

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Step 4.1.1.5.1

Multiply the exponents in ${(a^{\frac{1}{6}})}^{6}$ .

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Step 4.1.1.5.1.1

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

$\frac{a^{\frac{1}{6} \cdot 6}}{4^{6}} = 2^{- 6}$

Step 4.1.1.5.1.2

Cancel the common factor of $6$ .

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Step 4.1.1.5.1.2.1

Cancel the common factor.

$\frac{a^{\frac{1}{6} \cdot 6}}{4^{6}} = 2^{- 6}$

Step 4.1.1.5.1.2.2

Rewrite the expression.

$\frac{a^{1}}{4^{6}} = 2^{- 6}$

Step 4.1.1.5.2

Simplify.

$\frac{a}{4^{6}} = 2^{- 6}$

Step 4.1.1.6

Raise $4$ to the power of $6$ .

$\frac{a}{4096} = 2^{- 6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2^{- 6}$ .

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Step 4.2.1.1

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

$\frac{a}{4096} = \frac{1}{2^{6}}$

Step 4.2.1.2

Raise $2$ to the power of $6$ .

$\frac{a}{4096} = \frac{1}{64}$

Step 5

Solve for $a$ .

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Step 5.1

Multiply both sides of the equation by $4096$ .

$4096 \frac{a}{4096} = 4096 (\frac{1}{64})$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $4096$ .

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Step 5.2.1.1.1

Cancel the common factor.

$4096 \frac{a}{4096} = 4096 (\frac{1}{64})$

Step 5.2.1.1.2

Rewrite the expression.

$a = 4096 (\frac{1}{64})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Cancel the common factor of $64$ .

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Step 5.2.2.1.1

Factor $64$ out of $4096$ .

$a = 64 (64) \frac{1}{64}$

Step 5.2.2.1.2

Cancel the common factor.

$a = 64 \cdot 64 \frac{1}{64}$

Step 5.2.2.1.3

Rewrite the expression.

$a = 64$