Algebra Examples

$16^{2 x - 3} \cdot 4^{- 2 x} = 16$

Step 1

Rewrite $16$ as $2^{4}$ .

${(2^{4})}^{2 x - 3} \cdot 4^{- 2 x} = 16$

Step 2

Multiply the exponents in ${(2^{4})}^{2 x - 3}$ .

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Step 2.1

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

$2^{4 (2 x - 3)} \cdot 4^{- 2 x} = 16$

Step 2.2

Apply the distributive property.

$2^{4 (2 x) + 4 \cdot - 3} \cdot 4^{- 2 x} = 16$

Step 2.3

Multiply $2$ by $4$ .

$2^{8 x + 4 \cdot - 3} \cdot 4^{- 2 x} = 16$

Step 2.4

Multiply $4$ by $- 3$ .

$2^{8 x - 12} \cdot 4^{- 2 x} = 16$

Step 3

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

$2^{8 x - 12} \cdot {(2^{2})}^{- 2 x} = 16$

Step 4

Multiply the exponents in ${(2^{2})}^{- 2 x}$ .

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Step 4.1

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

$2^{8 x - 12} \cdot 2^{2 (- 2 x)} = 16$

Step 4.2

Multiply $- 2$ by $2$ .

$2^{8 x - 12} \cdot 2^{- 4 x} = 16$

Step 5

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

$2^{8 x - 12 - 4 x} = 16$

Step 6

Subtract $4 x$ from $8 x$ .

$2^{4 x - 12} = 16$

Step 7

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

$2^{4 x - 12} = 2^{4}$

Step 8

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

$4 x - 12 = 4$

Step 9

Solve for $x$ .

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Step 9.1

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

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Step 9.1.1

Add $12$ to both sides of the equation.

$4 x = 4 + 12$

Step 9.1.2

Add $4$ and $12$ .

$4 x = 16$

Step 9.2

Divide each term in $4 x = 16$ by $4$ and simplify.

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Step 9.2.1

Divide each term in $4 x = 16$ by $4$ .

$\frac{4 x}{4} = \frac{16}{4}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $4$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{16}{4}$

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{4}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $16$ by $4$ .

$x = 4$