College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Which choices are equivalent to the exponential expression below? Check all that apply.

[tex]\left(\frac{7}{12}\right)^2[/tex]

A. [tex]\frac{14}{24}[/tex]

B. [tex]2 \cdot\left(\frac{7}{12}\right)[/tex]

C. [tex]\left(\frac{7}{12}\right) \cdot\left(\frac{7}{12}\right)[/tex]

D. [tex]\frac{49}{144}[/tex]

E. [tex]\frac{7^2}{12^2}[/tex]

F. [tex]\frac{45}{12}[/tex]

Answer :

To determine which choices are equivalent to the exponential expression [tex]\(\left(\frac{7}{12}\right)^2\)[/tex], let's go through each option step-by-step:

1. Option A: [tex]\(\frac{14}{24}\)[/tex]
- Simplify [tex]\(\frac{14}{24}\)[/tex]. By dividing both the numerator and the denominator by 2, we get [tex]\(\frac{7}{12}\)[/tex].
- The original expression [tex]\(\left(\frac{7}{12}\right)^2\)[/tex] represents [tex]\(\frac{7}{12} \times \frac{7}{12}\)[/tex], not just [tex]\(\frac{7}{12}\)[/tex].
- Therefore, [tex]\(\frac{14}{24}\)[/tex] is not equivalent.

2. Option B: [tex]\(2 \cdot\left(\frac{7}{12}\right)\)[/tex]
- Calculate this expression: [tex]\(2 \cdot\frac{7}{12} = \frac{14}{12}\)[/tex].
- Simplifying [tex]\(\frac{14}{12}\)[/tex] gives [tex]\(\frac{7}{6}\)[/tex], which is not equivalent to [tex]\(\left(\frac{7}{12}\right)^2\)[/tex].
- So, this option is not equivalent.

3. Option C: [tex]\(\left(\frac{7}{12}\right) \cdot\left(\frac{7}{12}\right)\)[/tex]
- This is exactly the same as [tex]\(\left(\frac{7}{12}\right)^2\)[/tex].
- Thus, this option is equivalent.

4. Option D: [tex]\(\frac{49}{144}\)[/tex]
- Calculate [tex]\(\left(\frac{7}{12}\right)^2\)[/tex]:
- [tex]\(7^2 = 49\)[/tex]
- [tex]\(12^2 = 144\)[/tex]
- Therefore, [tex]\(\left(\frac{7}{12}\right)^2 = \frac{49}{144}\)[/tex].
- This option is equivalent.

5. Option E: [tex]\(\frac{7^2}{12^2}\)[/tex]
- This expression is another way of writing [tex]\(\left(\frac{7}{12}\right)^2\)[/tex].
- Since [tex]\(\left(\frac{7}{12}\right)^2 = \frac{49}{144}\)[/tex], this option is equivalent.

6. Option F: [tex]\(\frac{45}{12}\)[/tex]
- Simplify [tex]\(\frac{45}{12}\)[/tex] by dividing both the numerator and the denominator by 3, we get [tex]\(\frac{15}{4}\)[/tex].
- This does not match [tex]\(\left(\frac{7}{12}\right)^2\)[/tex].
- So, this option is not equivalent.

In summary, the choices that are equivalent to [tex]\(\left(\frac{7}{12}\right)^2\)[/tex] are:
- Option C: [tex]\(\left(\frac{7}{12}\right) \cdot\left(\frac{7}{12}\right)\)[/tex]
- Option D: [tex]\(\frac{49}{144}\)[/tex]
- Option E: [tex]\(\frac{7^2}{12^2}\)[/tex]