Precalculus Examples

${(\sqrt{3})}^{x + 1} = 9^{x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

${(3^{\frac{1}{2}})}^{x + 1} = 9^{x}$

Step 2

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

$3^{\frac{1}{2} (x + 1)} = 9^{x}$

Step 3

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

$3^{\frac{1}{2} (x + 1)} = 3^{2 x}$

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

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

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

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

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

Step 5.1.5

Multiply $\frac{1}{2}$ by $1$ .

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

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.

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Combine the numerators over the common denominator.

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

Step 5.2.5

Combine the numerators over the common denominator.

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

Step 5.2.6

Multiply $2$ by $- 2$ .

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

Step 5.2.7

Subtract $4 x$ from $x$ .

$\frac{- 3 x + 1}{2} = 0$

Step 5.2.8

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 1}{2} = 0$

Step 5.2.9

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (3 x) - 1 (- 1)}{2} = 0$

Step 5.2.10

Factor $- 1$ out of $- (3 x) - 1 (- 1)$ .

$\frac{- (3 x - 1)}{2} = 0$

Step 5.2.11

Rewrite $- (3 x - 1)$ as $- 1 (3 x - 1)$ .

$\frac{- 1 (3 x - 1)}{2} = 0$

Step 5.2.12

Move the negative in front of the fraction.

$- \frac{3 x - 1}{2} = 0$

Step 5.3

Set the numerator equal to zero.

$3 x - 1 = 0$

Step 5.4

Solve the equation for $x$ .

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Step 5.4.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 5.4.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 5.4.2.1

Divide each term in $3 x = 1$ by $3$ .

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

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

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

Step 5.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}$