To determine which expressions are equivalent to [tex]\(\frac{(5^{-3})^4}{5^6}\)[/tex], let's go through a detailed, step-by-step solution.
1. Simplify the numerator:
The expression to simplify is [tex]\((5^{-3})^4\)[/tex].
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ (5^{-3})^4 = 5^{-3 \cdot 4} = 5^{-12} \][/tex]
So, the expression now is:
[tex]\[ \frac{5^{-12}}{5^6} \][/tex]
2. Simplify the quotient:
To simplify [tex]\(\frac{5^{-12}}{5^6}\)[/tex], we use the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
Applying this rule, we have:
[tex]\[ \frac{5^{-12}}{5^6} = 5^{-12-6} = 5^{-18} \][/tex]
Now we need to check which of the given expressions are equivalent to [tex]\(5^{-18}\)[/tex].
Assessing each provided expression:
- [tex]\(\frac{5^1}{5^6}\)[/tex]:
Simplify using [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[ \frac{5^1}{5^6} = 5^{1-6} = 5^{-5} \][/tex]
This is not equivalent to [tex]\(5^{-18}\)[/tex].
- [tex]\(\frac{5^{-7}}{5^6}\)[/tex]:
Simplify using [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[ \frac{5^{-7}}{5^6} = 5^{-7-6} = 5^{-13} \][/tex]
This is not equivalent to [tex]\(5^{-18}\)[/tex].
- [tex]\(\frac{5^{-12}}{5^6}\)[/tex]:
Simplify using [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[ \frac{5^{-12}}{5^6} = 5^{-12-6} = 5^{-18} \][/tex]
This is equivalent to [tex]\(5^{-18}\)[/tex].
So, the only expression that is equivalent to [tex]\(\frac{(5^{-3})^4}{5^6}\)[/tex] is [tex]\(\frac{5^{-12}}{5^6}\)[/tex].
