Pre-Algebra Examples

$4 \sqrt{x + 7} = 23$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${(4 \sqrt{x + 7})}^{2} = 23^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

${(4 {(x + 7)}^{\frac{1}{2}})}^{2} = 23^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(4 {(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $4 {(x + 7)}^{\frac{1}{2}}$ .

$4^{2} {({(x + 7)}^{\frac{1}{2}})}^{2} = 23^{2}$

Step 2.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(x + 7)}^{\frac{1}{2}})}^{2} = 23^{2}$

Step 2.2.1.3

Multiply the exponents in ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

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

$16 {(x + 7)}^{\frac{1}{2} \cdot 2} = 23^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$16 {(x + 7)}^{\frac{1}{2} \cdot 2} = 23^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$16 {(x + 7)}^{1} = 23^{2}$

Step 2.2.1.4

Simplify.

$16 (x + 7) = 23^{2}$

Step 2.2.1.5

Apply the distributive property.

$16 x + 16 \cdot 7 = 23^{2}$

Step 2.2.1.6

Multiply $16$ by $7$ .

$16 x + 112 = 23^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $23$ to the power of $2$ .

$16 x + 112 = 529$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $112$ from both sides of the equation.

$16 x = 529 - 112$

Step 3.1.2

Subtract $112$ from $529$ .

$16 x = 417$

Step 3.2

Divide each term in $16 x = 417$ by $16$ and simplify.

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Step 3.2.1

Divide each term in $16 x = 417$ by $16$ .

$\frac{16 x}{16} = \frac{417}{16}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $16$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{417}{16}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{417}{16}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{417}{16}$

Decimal Form:

$x = 26.0625$

Mixed Number Form:

$x = 26 \frac{1}{16}$