Question

question 6 of 10
which expressions are equivalent to the one below? check all that apply.
$49^x$

a. $7^{2x}$
b. $7^x \cdot 7^x$
c. $(7 \cdot 7)^x$
d. $7 \cdot 7^x$
e. $7^2 \cdot 7^x$
f. $7 \cdot 7^{2x}$

Explanation:

Step1: Recall exponent rules

We know that \(a^{m}\cdot a^{n}=a^{m + n}\) and \((a^{m})^{n}=a^{mn}\), also \((ab)^{n}=a^{n}b^{n}\). Given the expression \(49^{x}\), and since \(49 = 7\times7=7^{2}\).

Step2: Analyze Option A

For option A: \(7^{2x}\). Using the rule \((a^{m})^{n}=a^{mn}\), we can write \(7^{2x}=(7^{2})^{x}\). Since \(7^{2} = 49\), then \((7^{2})^{x}=49^{x}\). So option A is equivalent.

Step3: Analyze Option B

For option B: \(7^{x}\cdot7^{x}\). Using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(7^{x}\cdot7^{x}=7^{x + x}=7^{2x}=(7^{2})^{x}=49^{x}\). So option B is equivalent.

Step4: Analyze Option C

For option C: \((7\cdot7)^{x}\). Since \(7\times7 = 49\), then \((7\times7)^{x}=49^{x}\). So option C is equivalent.

Step5: Analyze Option D

For option D: \(7\cdot7^{x}\). Using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), we get \(7\cdot7^{x}=7^{1 + x}
eq49^{x}\) (since \(49^{x}=7^{2x}\) and \(1 + x
eq2x\) for \(x
eq1\)). So option D is not equivalent.

Step6: Analyze Option E

For option E: \(7^{2}\cdot7^{x}\). Using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(7^{2}\cdot7^{x}=7^{2 + x}
eq49^{x}\) (since \(49^{x}=7^{2x}\) and \(2 + x
eq2x\) for \(x
eq2\)). So option E is not equivalent.

Step7: Analyze Option F

For option F: \(7\cdot7^{2x}\). Using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), we get \(7\cdot7^{2x}=7^{1+2x}
eq49^{x}\) (since \(49^{x}=7^{2x}\) and \(1 + 2x
eq2x\)). So option F is not equivalent.

Answer:

A. \(7^{2x}\), B. \(7^{x}\cdot7^{x}\), C. \((7\cdot7)^{x}\)