Question

solving exponential equations by rewriting the base if $8^y = 16^{y + 2}$, what is the value of $y$? -1 -2 -8 -4

Explanation:

Step1: Rewrite bases as powers of 2

We know that \(8 = 2^3\) and \(16 = 2^4\). So we can rewrite the equation \(8^y=16^{y + 2}\) as \((2^3)^y=(2^4)^{y + 2}\).

Step2: Apply exponent rules

Using the exponent rule \((a^m)^n=a^{mn}\), we get \(2^{3y}=2^{4(y + 2)}\).

Step3: Set exponents equal

Since the bases are the same and the equation holds, we can set the exponents equal to each other: \(3y=4(y + 2)\).

Step4: Solve for y

Expand the right side: \(3y = 4y+8\).
Subtract \(4y\) from both sides: \(3y-4y=4y + 8-4y\), which simplifies to \(-y = 8\).
Multiply both sides by - 1: \(y=-8\).

Answer:

\(-8\)