To determine which of the given expressions is equal to [tex]\(\frac{5^6}{5^2}\)[/tex], we need to simplify [tex]\(\frac{5^6}{5^2}\)[/tex] using the properties of exponents.
When dividing exponential terms with the same base, we subtract the exponent of the denominator from the exponent of the numerator. So:
[tex]\[
\frac{5^6}{5^2} = 5^{6-2} = 5^4
\][/tex]
Now we need to check which of the given expressions is equal to [tex]\(5^4\)[/tex].
Let's evaluate each expression:
1. [tex]\(5 \cdot 5 \cdot 5\)[/tex]
[tex]\[
5 \cdot 5 \cdot 5 = 5^3
\][/tex]
This expression equals [tex]\(5^3\)[/tex], which is not equal to [tex]\(5^4\)[/tex].
2. [tex]\(\overline{5 \cdot 5 \cdot 5 \cdot 5}\)[/tex]
[tex]\[
5 \cdot 5 \cdot 5 \cdot 5 = 5^4
\][/tex]
This expression equals [tex]\(5^4\)[/tex], which matches our simplified result.
3. [tex]\(5 \cdot 5 \cdot 5 \cdot 5\)[/tex]
[tex]\[
5 \cdot 5 \cdot 5 \cdot 5 = 5^4
\][/tex]
This expression also equals [tex]\(5^4\)[/tex], which matches our simplified result.
4. The number [tex]\(3\)[/tex]
[tex]\[
3 \neq 5^4
\][/tex]
This number is not equal to [tex]\(5^4\)[/tex].
Given the results:
- [tex]\(5^4 = \overline{5 \cdot 5 \cdot 5 \cdot 5}\)[/tex]
- [tex]\(5^4 = 5 \cdot 5 \cdot 5 \cdot 5\)[/tex]
Both the second and third expressions are equal to [tex]\(5^4\)[/tex], but seeing as the second representation uses discrepancy notation, we will prefer the third option as a clean match.
Thus, the expression [tex]\( 5 \cdot 5 \cdot 5 \cdot 5 \)[/tex] is equal to [tex]\(\frac{5^6}{5^2} \)[/tex].
