Question

name: date: pd: 7.3c solving exponential functions quick write the function shown is solved incorrectly. explain how you know that the function is solved incorrectly using complete sentences and mathematical vocabulary. $4^{2x - 6}$ $4(2x - 6) = 64$ $8x - 24 = 64$ $+24 -24$ $\frac{8x}{8} = \frac{88}{8}$ $x = 11$

Explanation:

1. The original expression is an exponential function $4^{2x-6}$, not a linear expression. The solver incorrectly treated the exponential base and exponent as a linear product $4(2x-6)$, which violates exponential function rules.
2. Even in the incorrect linear expansion, $4(2x-6)$ simplifies to $8x - 24$, not $5x - 24$. This is an arithmetic error in distributing the coefficient.
3. When solving $8x - 24 = 64$, adding 24 to both sides gives $8x = 88$, not $8x = 58$. This is an arithmetic mistake in integer addition.

Answer:

The solution is incorrect for multiple reasons:

1. The core error is misinterpreting the exponential expression $4^{2x-6}$ as a linear product $4(2x-6)$. Exponential expressions have a base raised to an exponent, which cannot be rewritten as a simple multiplication of the base and exponent.
2. There is an arithmetic error in expanding the incorrect linear term: $4(2x-6) = 8x - 24$, not $5x - 24$.
3. There is a second arithmetic error when isolating the variable: adding 24 to both sides of $8x - 24 = 64$ results in $8x = 88$, not $8x = 58$, so the final value of $x$ is also wrong.