Question

question 10 of 10
which expressions are equivalent to the one below? check all that apply.
$4^3 \cdot 4^x$

a. $64 \cdot 4^x$
b. $4^{3 - x}$
c. $4^{3 + x}$
d. $16^{3x}$
e. $4^{3x}$
f. $(4 \cdot x)^3$

Explanation:

Step1: Recall exponent rule

When multiplying exponents with the same base, \(a^m \cdot a^n = a^{m + n}\). So for \(4^3 \cdot 4^x\), we can apply this rule.
\(4^3 \cdot 4^x = 4^{3 + x}\)

Step2: Evaluate \(4^3\)

Calculate \(4^3\): \(4^3=4\times4\times4 = 64\). So \(4^3 \cdot 4^x=64\cdot4^x\)

Step3: Analyze each option

• Option A: \(64\cdot4^x\) matches \(4^3 \cdot 4^x\) (since \(4^3 = 64\)), so A is equivalent.
• Option B: \(4^{3 - x}\) is not equivalent (subtraction in exponent, not addition).
• Option C: \(4^{3 + x}\) matches the result from the exponent rule, so C is equivalent.
• Option D: \(16^{3x}\) has base 16 and different exponent structure, not equivalent.
• Option E: \(4^{3x}\) has exponent \(3x\) (multiplication), not \(3 + x\), not equivalent.
• Option F: \((4\cdot x)^3=4^3\cdot x^3 = 64x^3\), not equivalent to \(4^3 \cdot 4^x\).

Answer:

A. \(64 \cdot 4^x\), C. \(4^{3 + x}\)