Questions: If (32^2c=8^c+7), what is the value of (c)? 1 2 3 5

Solution Steps

We start by expressing both sides of the equation \( 32^{2c} = 8^{c+7} \) with the same base. Since \( 32 \) and \( 8 \) are both powers of \( 2 \), we can rewrite them as: \[ 32 = 2^5 \quad \text{and} \quad 8 = 2^3. \] Substituting these into the equation gives: \[ (2^5)^{2c} = (2^3)^{c+7}. \]

Using the power of a power property \((a^m)^n = a^{mn}\), we simplify the exponents: \[ 2^{5 \cdot 2c} = 2^{3 \cdot (c+7)}. \] This simplifies to: \[ 2^{10c} = 2^{3c + 21}. \]

Since the bases are the same, we can set the exponents equal to each other: \[ 10c = 3c + 21. \]

Subtract \( 3c \) from both sides: \[ 10c - 3c = 21, \] which simplifies to: \[ 7c = 21. \] Divide both sides by \( 7 \): \[ c = 3. \]

Final Answer