Answer :
To solve the given expression \(\left(6^{-2} \cdot 6^5\right)^{-3}\), we can follow these steps:
1. Combine the Bases with Exponents:
- We start with the expression inside the parentheses: \(6^{-2} \cdot 6^5\).
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 6^{-2} \cdot 6^5 = 6^{-2 + 5} = 6^3 \][/tex]
2. Raise the Result to the Power of \(-3\):
- Now we need to raise \(6^3\) to the power of \(-3\):
[tex]\[ (6^3)^{-3} \][/tex]
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\):
[tex]\[ (6^3)^{-3} = 6^{3 \cdot (-3)} = 6^{-9} \][/tex]
3. Simplify the Expression:
- An exponent with a negative power can be written as a fraction:
[tex]\[ 6^{-9} = \frac{1}{6^9} \][/tex]
Thus, the expression \(\left(6^{-2} \cdot 6^5\right)^{-3}\) simplifies to \(\frac{1}{6^9}\).
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{1}{6^9}} \][/tex]
1. Combine the Bases with Exponents:
- We start with the expression inside the parentheses: \(6^{-2} \cdot 6^5\).
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 6^{-2} \cdot 6^5 = 6^{-2 + 5} = 6^3 \][/tex]
2. Raise the Result to the Power of \(-3\):
- Now we need to raise \(6^3\) to the power of \(-3\):
[tex]\[ (6^3)^{-3} \][/tex]
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\):
[tex]\[ (6^3)^{-3} = 6^{3 \cdot (-3)} = 6^{-9} \][/tex]
3. Simplify the Expression:
- An exponent with a negative power can be written as a fraction:
[tex]\[ 6^{-9} = \frac{1}{6^9} \][/tex]
Thus, the expression \(\left(6^{-2} \cdot 6^5\right)^{-3}\) simplifies to \(\frac{1}{6^9}\).
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{1}{6^9}} \][/tex]
