Algebra Examples

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

Step 1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2

One to any power is one.

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

Step 3

Move $2^{2 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- 2 x} = 2^{3 - x}$

Step 4

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

$- 2 x = 3 - x$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Add $x$ to both sides of the equation.

$- 2 x + x = 3$

Step 5.1.2

Add $- 2 x$ and $x$ .

$- x = 3$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $3$ by $- 1$ .

$x = - 3$