Precalculus Examples

$4^{x} \cdot {(\frac{1}{2})}^{3 - 2 x} = 8 \cdot {(2^{x})}^{2}$

Step 1

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

$4^{x} \cdot {(\frac{1}{2})}^{3 - 2 x} = 8 \cdot 2^{x \cdot 2}$

Step 2

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

$4^{x} \cdot \frac{1^{3 - 2 x}}{2^{3 - 2 x}} = 8 \cdot 2^{x \cdot 2}$

Step 3

One to any power is one.

$4^{x} \cdot \frac{1}{2^{3 - 2 x}} = 8 \cdot 2^{x \cdot 2}$

Step 4

Move $2^{3 - 2 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{x} \cdot 2^{- (3 - 2 x)} = 8 \cdot 2^{x \cdot 2}$

Step 5

Rewrite $4$ as $2^{2}$ .

${(2^{2})}^{x} \cdot 2^{- (3 - 2 x)} = 8 \cdot 2^{x \cdot 2}$

Step 6

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

$2^{2 x} \cdot 2^{- (3 - 2 x)} = 8 \cdot 2^{x \cdot 2}$

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{2 x - (3 - 2 x)} = 8 \cdot 2^{x \cdot 2}$

Step 8

Rewrite $8$ as $2^{3}$ .

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

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10

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

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

Step 11

Solve for $x$ .

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Step 11.1

Simplify $2 x - (3 - 2 x)$ .

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Step 11.1.1

Simplify each term.

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Step 11.1.1.1

Apply the distributive property.

$2 x - 1 \cdot 3 - (- 2 x) = 3 + x \cdot 2$

Step 11.1.1.2

Multiply $- 1$ by $3$ .

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

Step 11.1.1.3

Multiply $- 2$ by $- 1$ .

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

Step 11.1.2

Add $2 x$ and $2 x$ .

$4 x - 3 = 3 + x \cdot 2$

Step 11.2

Move $2$ to the left of $x$ .

$4 x - 3 = 3 + 2 x$

Step 11.3

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

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Step 11.3.1

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

$4 x - 3 - 2 x = 3$

Step 11.3.2

Subtract $2 x$ from $4 x$ .

$2 x - 3 = 3$

Step 11.4

Move all terms not containing $x$ to the right side of the equation.

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Step 11.4.1

Add $3$ to both sides of the equation.

$2 x = 3 + 3$

Step 11.4.2

Add $3$ and $3$ .

$2 x = 6$

Step 11.5

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 11.5.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 11.5.2

Simplify the left side.

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Step 11.5.2.1

Cancel the common factor of $2$ .

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Step 11.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 11.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 11.5.3

Simplify the right side.

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Step 11.5.3.1

Divide $6$ by $2$ .

$x = 3$