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]
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]
