Precalculus Examples

${(\sqrt{3})}^{x + 5} = 81^{x}$

Step 1

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

${(3^{\frac{1}{2}})}^{x + 5} = 81^{x}$

Step 2

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

$3^{\frac{1}{2} (x + 5)} = 81^{x}$

Step 3

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

$3^{\frac{1}{2} (x + 5)} = 3^{4 x}$

Step 4

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

$\frac{1}{2} (x + 5) = 4 x$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Simplify $\frac{1}{2} \cdot (x + 5)$ .

Tap for more steps...

Step 5.1.1

Rewrite.

$0 + 0 + \frac{1}{2} \cdot (x + 5) = 4 x$

Step 5.1.2

Simplify by adding zeros.

$\frac{1}{2} \cdot (x + 5) = 4 x$

Step 5.1.3

Apply the distributive property.

$\frac{1}{2} x + \frac{1}{2} \cdot 5 = 4 x$

Step 5.1.4

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

$\frac{x}{2} + \frac{1}{2} \cdot 5 = 4 x$

Step 5.1.5

Combine $\frac{1}{2}$ and $5$ .

$\frac{x}{2} + \frac{5}{2} = 4 x$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $4 x$ from both sides of the equation.

$\frac{x}{2} + \frac{5}{2} - 4 x = 0$

Step 5.2.2

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} - 4 x \cdot \frac{2}{2} + \frac{5}{2} = 0$

Step 5.2.3

Combine $- 4 x$ and $\frac{2}{2}$ .

$\frac{x}{2} + \frac{- 4 x \cdot 2}{2} + \frac{5}{2} = 0$

Step 5.2.4

Combine the numerators over the common denominator.

$\frac{x - 4 x \cdot 2}{2} + \frac{5}{2} = 0$

Step 5.2.5

Combine the numerators over the common denominator.

$\frac{x - 4 x \cdot 2 + 5}{2} = 0$

Step 5.2.6

Multiply $2$ by $- 4$ .

$\frac{x - 8 x + 5}{2} = 0$

Step 5.2.7

Subtract $8 x$ from $x$ .

$\frac{- 7 x + 5}{2} = 0$

Step 5.2.8

Factor $- 1$ out of $- 7 x$ .

$\frac{- (7 x) + 5}{2} = 0$

Step 5.2.9

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (7 x) - 1 (- 5)}{2} = 0$

Step 5.2.10

Factor $- 1$ out of $- (7 x) - 1 (- 5)$ .

$\frac{- (7 x - 5)}{2} = 0$

Step 5.2.11

Rewrite $- (7 x - 5)$ as $- 1 (7 x - 5)$ .

$\frac{- 1 (7 x - 5)}{2} = 0$

Step 5.2.12

Move the negative in front of the fraction.

$- \frac{7 x - 5}{2} = 0$

Step 5.3

Set the numerator equal to zero.

$7 x - 5 = 0$

Step 5.4

Solve the equation for $x$ .

Tap for more steps...

Step 5.4.1

Add $5$ to both sides of the equation.

$7 x = 5$

Step 5.4.2

Divide each term in $7 x = 5$ by $7$ and simplify.

Tap for more steps...

Step 5.4.2.1

Divide each term in $7 x = 5$ by $7$ .

$\frac{7 x}{7} = \frac{5}{7}$

Step 5.4.2.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{5}{7}$

Step 5.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{7}$

Decimal Form:

$x = 0.\overline{714285}$