Precalculus Examples

$125^{3 k} = 625^{2 k + 3}$

Step 1

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

$5^{3 (3 k)} = 5^{4 (2 k + 3)}$

Step 2

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

$3 (3 k) = 4 (2 k + 3)$

Step 3

Solve for $k$ .

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Step 3.1

Multiply $3$ by $3$ .

$9 k = 4 (2 k + 3)$

Step 3.2

Simplify $4 (2 k + 3)$ .

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Step 3.2.1

Apply the distributive property.

$9 k = 4 (2 k) + 4 \cdot 3$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $2$ by $4$ .

$9 k = 8 k + 4 \cdot 3$

Step 3.2.2.2

Multiply $4$ by $3$ .

$9 k = 8 k + 12$

Step 3.3

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

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Step 3.3.1

Subtract $8 k$ from both sides of the equation.

$9 k - 8 k = 12$

Step 3.3.2

Subtract $8 k$ from $9 k$ .

$k = 12$