Algebra Examples

$h (q) = \sqrt[4]{4 q + 11}$ ; solve for $h (q) = 1$

Step 1

Substitute $1$ for $h (q)$ .

$1 = \sqrt[4]{4 q + 11}$

Step 2

Solve for $q$ .

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Step 2.1

Rewrite the equation as $\sqrt[4]{4 q + 11} = 1$ .

$\sqrt[4]{4 q + 11} = 1$

Step 2.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{4 q + 11}}^{4} = 1^{4}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{4 q + 11}$ as ${(4 q + 11)}^{\frac{1}{4}}$ .

${({(4 q + 11)}^{\frac{1}{4}})}^{4} = 1^{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(4 q + 11)}^{\frac{1}{4}})}^{4}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(4 q + 11)}^{\frac{1}{4}})}^{4}$ .

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Step 2.3.2.1.1.1

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

${(4 q + 11)}^{\frac{1}{4} \cdot 4} = 1^{4}$

Step 2.3.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(4 q + 11)}^{\frac{1}{4} \cdot 4} = 1^{4}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(4 q + 11)}^{1} = 1^{4}$

Step 2.3.2.1.2

Simplify.

$4 q + 11 = 1^{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

One to any power is one.

$4 q + 11 = 1$

Step 2.4

Solve for $q$ .

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Step 2.4.1

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

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Step 2.4.1.1

Subtract $11$ from both sides of the equation.

$4 q = 1 - 11$

Step 2.4.1.2

Subtract $11$ from $1$ .

$4 q = - 10$

Step 2.4.2

Divide each term in $4 q = - 10$ by $4$ and simplify.

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Step 2.4.2.1

Divide each term in $4 q = - 10$ by $4$ .

$\frac{4 q}{4} = \frac{- 10}{4}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{4 q}{4} = \frac{- 10}{4}$

Step 2.4.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{- 10}{4}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Cancel the common factor of $- 10$ and $4$ .

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Step 2.4.2.3.1.1

Factor $2$ out of $- 10$ .

$q = \frac{2 (- 5)}{4}$

Step 2.4.2.3.1.2

Cancel the common factors.

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Step 2.4.2.3.1.2.1

Factor $2$ out of $4$ .

$q = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 2.4.2.3.1.2.2

Cancel the common factor.

$q = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 2.4.2.3.1.2.3

Rewrite the expression.

$q = \frac{- 5}{2}$

Step 2.4.2.3.2

Move the negative in front of the fraction.

$q = - \frac{5}{2}$