Algebra Examples

$16^{3 x} = (2^{5 x + 2}) (8^{2 x})$

Step 1

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

$16^{3 x} = 2^{5 x + 2} {(2^{3})}^{2 x}$

Step 2

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

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Step 2.1

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

$16^{3 x} = 2^{5 x + 2} \cdot 2^{3 (2 x)}$

Step 2.2

Multiply $2$ by $3$ .

$16^{3 x} = 2^{5 x + 2} \cdot 2^{6 x}$

Step 3

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

$16^{3 x} = 2^{5 x + 2 + 6 x}$

Step 4

Add $5 x$ and $6 x$ .

$16^{3 x} = 2^{11 x + 2}$

Step 5

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

$2^{4 (3 x)} = 2^{11 x + 2}$

Step 6

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

$4 (3 x) = 11 x + 2$

Step 7

Solve for $x$ .

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Step 7.1

Multiply $3$ by $4$ .

$12 x = 11 x + 2$

Step 7.2

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

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Step 7.2.1

Subtract $11 x$ from both sides of the equation.

$12 x - 11 x = 2$

Step 7.2.2

Subtract $11 x$ from $12 x$ .

$x = 2$