Solve the exponential equation by using the property that bx = by implies x = y, for b 0 and b≠ 1. $$8 ^ { 2 x } = 64 ^ { x ^ { 2 } }$$
A) $\{ 0 \}$
B) $\{ 64 \}$
C) $\{ 0,1 \}$
D) $\left\{ \frac { 1 } { 2 } , - \frac { 1 } { 3 } \right\}$

Question

Solve the exponential equation by using the property that bx = by implies x = y, for b > 0 and b≠ 1. $$8 ^ { 2 x } = 64 ^ { x ^ { 2 } }$$
A) $\{ 0 \}$
B) $\{ 64 \}$
C) $\{ 0,1 \}$
D) $\left\{ \frac { 1 } { 2 } , - \frac { 1 } { 3 } \right\}$

Quizplus · Accepted Answer

First, express both sides of the equation with a common base. The number 8 can be written as $2^3$ and 64 as $2^6$. Thus, the equation becomes $(2^3)^{2x} = (2^6)^{x^2}$. Simplifying the exponents gives $2^{6x} = 2^{6x^2}$. By the property that $b^x = b^y$ implies $x = y$, we have $6x = 6x^2$. Dividing both sides by 6 gives $x = x^2$. Rearranging gives $x^2 - x = 0$, which factors to $x(x - 1) = 0$. This yields the solutions $x = 0$ and $x = 1$.

Solve the Exponential Equation by Using the Property That Bx 82x=64x28 ^ { 2 x } = 64 ^ { x ^ { 2 } }82x=64x2

Solve the Exponential Equation by Using the Property That Bx $8 ^ { 2 x } = 64 ^ { x ^ { 2 } }$