Question

2.3c 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 reasoning.
$4^{2x - 6}$
$4(2x - 6) = 64$
$8x - 24 = 64$
$ + 24 \\ \\ \\ \\ + 24$
$\frac{8x}{8} = \frac{88}{8}$
$x = 11$

Explanation:

Step1: Identify original exponential term

The given expression is $4^{2x-6}$, and it was incorrectly treated as a linear expression.

Step2: Correct base conversion

First, rewrite 64 as a power of 4: $64 = 4^3$.

Step3: Set exponents equal

For $4^{2x-6}=4^3$, equate exponents:
$2x - 6 = 3$

Step4: Solve for x

Add 6 to both sides:
$2x = 3 + 6 = 9$
Divide by 2:
$x = \frac{9}{2}$

Step5: Analyze errors in work

1. The original term $4^{2x-6}$ was incorrectly expanded as $4(2x-6)$ (treating exponent as multiplication, which violates exponential rules).
2. The expansion $4(2x-6)$ was miscalculated as $8x-24$ (this part is algebraically correct, but the initial step was wrong), then incorrectly rewritten as $5x-24$.
3. When solving $8x-24=64$, adding 24 to both sides gives $8x=88$, not $8x=58$, and the final $x=11$ is based on multiple prior errors.

Answer:

The work is incorrect because:

1. The exponential expression $4^{2x-6}$ was incorrectly treated as a linear product $4(2x-6)$—exponential notation means repeated multiplication of the base, not multiplying the base by the exponent.
2. There were arithmetic errors in expanding and solving the incorrect linear equation: $4(2x-6)$ was miswritten as $5x-24$, and solving $8x-24=64$ would give $8x=88$ (not $8x=58$), though this is irrelevant because the initial step violated exponential rules.
3. The correct solution for $4^{2x-6}=64$ is $x=\frac{9}{2}$, since we rewrite 64 as $4^3$, set $2x-6=3$, and solve for $x$.