Algebra Examples

${(\sqrt{n - 19})}^{4} = 16$

Step 1

Rewrite ${\sqrt{n - 19}}^{4}$ as ${(n - 19)}^{2}$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n - 19}$ as ${(n - 19)}^{\frac{1}{2}}$ .

${({(n - 19)}^{\frac{1}{2}})}^{4} = 16$

Step 1.2

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

${(n - 19)}^{\frac{1}{2} \cdot 4} = 16$

Step 1.3

Combine $\frac{1}{2}$ and $4$ .

${(n - 19)}^{\frac{4}{2}} = 16$

Step 1.4

Cancel the common factor of $4$ and $2$ .

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Step 1.4.1

Factor $2$ out of $4$ .

${(n - 19)}^{\frac{2 \cdot 2}{2}} = 16$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $2$ .

${(n - 19)}^{\frac{2 \cdot 2}{2 (1)}} = 16$

Step 1.4.2.2

Cancel the common factor.

${(n - 19)}^{\frac{2 \cdot 2}{2 \cdot 1}} = 16$

Step 1.4.2.3

Rewrite the expression.

${(n - 19)}^{\frac{2}{1}} = 16$

Step 1.4.2.4

Divide $2$ by $1$ .

${(n - 19)}^{2} = 16$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$n - 19 = \pm \sqrt{16}$

Step 3

Simplify $\pm \sqrt{16}$ .

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$n - 19 = \pm \sqrt{4^{2}}$

Step 3.2

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

$n - 19 = \pm 4$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$n - 19 = 4$

Step 4.2

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

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Step 4.2.1

Add $19$ to both sides of the equation.

$n = 4 + 19$

Step 4.2.2

Add $4$ and $19$ .

$n = 23$

Step 4.3

Next, use the negative value of the $\pm$ to find the second solution.

$n - 19 = - 4$

Step 4.4

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

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Step 4.4.1

Add $19$ to both sides of the equation.

$n = - 4 + 19$

Step 4.4.2

Add $- 4$ and $19$ .

$n = 15$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

$n = 23, 15$

Step 5

Exclude the solutions that do not make ${(\sqrt{n - 19})}^{4} = 16$ true.

$n = 23$