Algebra Examples

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

Step 1

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

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

Step 2

Apply the product rule to $\frac{1}{a}$ .

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

Step 3

One to any power is one.

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

Step 4

Move $a^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$a^{2 (x - 3)} = a^{- x}$

Step 5

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

$2 (x - 3) = - x$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $2 (x - 3)$ .

Tap for more steps...

Step 6.1.1

Rewrite.

$0 + 0 + 2 (x - 3) = - x$

Step 6.1.2

Simplify by adding zeros.

$2 (x - 3) = - x$

Step 6.1.3

Apply the distributive property.

$2 x + 2 \cdot - 3 = - x$

Step 6.1.4

Multiply $2$ by $- 3$ .

$2 x - 6 = - x$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Add $x$ to both sides of the equation.

$2 x - 6 + x = 0$

Step 6.2.2

Add $2 x$ and $x$ .

$3 x - 6 = 0$

Step 6.3

Add $6$ to both sides of the equation.

$3 x = 6$

Step 6.4

Divide each term in $3 x = 6$ by $3$ and simplify.

Tap for more steps...

Step 6.4.1

Divide each term in $3 x = 6$ by $3$ .

$\frac{3 x}{3} = \frac{6}{3}$

Step 6.4.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{6}{3}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{3}$

Step 6.4.3

Simplify the right side.

Tap for more steps...

Step 6.4.3.1

Divide $6$ by $3$ .

$x = 2$