Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Create equivalent expressions in the equation that all have equal bases.

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

Step 4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

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

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

$0 + 0 + \frac{1}{2} \cdot (x + 4) = 2 x$

Step 5.1.2

Simplify by adding zeros.

$\frac{1}{2} \cdot (x + 4) = 2 x$

Step 5.1.3

Apply the distributive property.

$\frac{1}{2} x + \frac{1}{2} \cdot 4 = 2 x$

Step 5.1.4

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

$\frac{x}{2} + \frac{1}{2} \cdot 4 = 2 x$

Step 5.1.5

Cancel the common factor of $2$ .

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Step 5.1.5.1

Factor $2$ out of $4$ .

$\frac{x}{2} + \frac{1}{2} \cdot (2 (2)) = 2 x$

Step 5.1.5.2

Cancel the common factor.

$\frac{x}{2} + \frac{1}{2} \cdot (2 \cdot 2) = 2 x$

Step 5.1.5.3

Rewrite the expression.

$\frac{x}{2} + 2 = 2 x$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

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Step 5.2.1

Subtract $2 x$ from both sides of the equation.

$\frac{x}{2} + 2 - 2 x = 0$

Step 5.2.2

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} - 2 x \cdot \frac{2}{2} + 2 = 0$

Step 5.2.3

Combine $- 2 x$ and $\frac{2}{2}$ .

$\frac{x}{2} + \frac{- 2 x \cdot 2}{2} + 2 = 0$

Step 5.2.4

Combine the numerators over the common denominator.

$\frac{x - 2 x \cdot 2}{2} + 2 = 0$

Step 5.2.5

Simplify each term.

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Step 5.2.5.1

Simplify the numerator.

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Step 5.2.5.1.1

Factor $x$ out of $x - 2 x \cdot 2$ .

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Step 5.2.5.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - 2 x \cdot 2}{2} + 2 = 0$

Step 5.2.5.1.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - 2 x \cdot 2}{2} + 2 = 0$

Step 5.2.5.1.1.3

Factor $x$ out of $- 2 x \cdot 2$ .

$\frac{x \cdot 1 + x (- 2 \cdot 2)}{2} + 2 = 0$

Step 5.2.5.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 2 \cdot 2)$ .

$\frac{x (1 - 2 \cdot 2)}{2} + 2 = 0$

Step 5.2.5.1.2

Multiply $- 2$ by $2$ .

$\frac{x (1 - 4)}{2} + 2 = 0$

Step 5.2.5.1.3

Subtract $4$ from $1$ .

$\frac{x \cdot - 3}{2} + 2 = 0$

Step 5.2.5.2

Move $- 3$ to the left of $x$ .

$\frac{- 3 \cdot x}{2} + 2 = 0$

Step 5.2.5.3

Move the negative in front of the fraction.

$- \frac{3 x}{2} + 2 = 0$

Step 5.3

Subtract $2$ from both sides of the equation.

$- \frac{3 x}{2} = - 2$

Step 5.4

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 2$

Step 5.5

Simplify both sides of the equation.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Simplify $- \frac{2}{3} (- \frac{3 x}{2})$ .

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Step 5.5.1.1.1

Cancel the common factor of $2$ .

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Step 5.5.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.1.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.2

Cancel the common factor of $3$ .

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Step 5.5.1.1.2.1

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.3

Multiply.

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Step 5.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{3} \cdot - 2$

Step 5.5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{3} \cdot - 2$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Multiply $- \frac{2}{3} \cdot - 2$ .

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Step 5.5.2.1.1

Multiply $- 2$ by $- 1$ .

$x = 2 (\frac{2}{3})$

Step 5.5.2.1.2

Combine $2$ and $\frac{2}{3}$ .

$x = \frac{2 \cdot 2}{3}$

Step 5.5.2.1.3

Multiply $2$ by $2$ .

$x = \frac{4}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}$

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

$x = 1 \frac{1}{3}$