To determine which of the given expressions are equivalent to [tex]\(7^{-2} \cdot 7^6\)[/tex], we can simplify the original expression step-by-step using the laws of exponents.
The original expression is:
[tex]\[ 7^{-2} \cdot 7^6 \][/tex]
Firstly, apply the product rule of exponents, which states that [tex]\(a^m \cdot a^n = a^{m + n}\)[/tex]:
[tex]\[ 7^{-2} \cdot 7^6 = 7^{-2 + 6} \][/tex]
Simplify the exponent:
[tex]\[ -2 + 6 = 4 \][/tex]
So, the expression simplifies to:
[tex]\[ 7^4 \][/tex]
Now, we need to check which of the given expressions are equivalent to [tex]\(7^4\)[/tex].
1. For [tex]\(\frac{7^2}{7^{-2}}\)[/tex]:
- Use the quotient rule of exponents, which states that [tex]\(\frac{a^m}{a^n} = a^{m - n}\)[/tex]:
[tex]\[ \frac{7^2}{7^{-2}} = 7^{2 - (-2)} = 7^{2 + 2} = 7^4 \][/tex]
- This expression is equivalent to [tex]\(7^4\)[/tex].
2. For [tex]\(\frac{7^6}{7^{-2}}\)[/tex]:
- Similarly, apply the quotient rule of exponents:
[tex]\[ \frac{7^6}{7^{-2}} = 7^{6 - (-2)} = 7^{6 + 2} = 7^8 \][/tex]
- This expression is equivalent to [tex]\(7^8\)[/tex], and not [tex]\(7^4\)[/tex].
3. For [tex]\(7^{-120}\)[/tex]:
- This expression is simply [tex]\(7^{-120}\)[/tex], which is clearly not equivalent to [tex]\(7^4\)[/tex].
4. For [tex]\(\left(7^2\right)^2\)[/tex]:
- Use the power rule of exponents, which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ \left(7^2\right)^2 = 7^{2 \cdot 2} = 7^4 \][/tex]
- This expression is equivalent to [tex]\(7^4\)[/tex].
Thus, the expressions that are equivalent to [tex]\(7^{-2} \cdot 7^6\)[/tex] are:
[tex]\[ \boxed{\frac{7^2}{7^{-2}} \text{ and } \left(7^2\right)^2} \][/tex]
These correspond to options (A) and (D).
