Question
Simplify using the negative integer rule of exponents. Leave your answer in exponential form with positive exponent six to the -3×6 to the fifth over six to the ninth.
Answers
Answered by
GPT-4o mini
To simplify the expression \(\frac{6^{-3} \cdot 6^{5}}{6^{9}}\), we can use the rules of exponents.
First, we can combine the exponents in the numerator using the product rule, which states that \(a^m \cdot a^n = a^{m+n}\):
\[
6^{-3} \cdot 6^{5} = 6^{-3 + 5} = 6^{2}
\]
Now the expression becomes:
\[
\frac{6^{2}}{6^{9}}
\]
Next, we can apply the quotient rule, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{6^{2}}{6^{9}} = 6^{2 - 9} = 6^{-7}
\]
To express this in exponential form with a positive exponent, we use the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^{n}}\):
\[
6^{-7} = \frac{1}{6^{7}}
\]
Thus, the final answer in exponential form with a positive exponent is:
\[
\frac{1}{6^{7}}
\]
First, we can combine the exponents in the numerator using the product rule, which states that \(a^m \cdot a^n = a^{m+n}\):
\[
6^{-3} \cdot 6^{5} = 6^{-3 + 5} = 6^{2}
\]
Now the expression becomes:
\[
\frac{6^{2}}{6^{9}}
\]
Next, we can apply the quotient rule, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{6^{2}}{6^{9}} = 6^{2 - 9} = 6^{-7}
\]
To express this in exponential form with a positive exponent, we use the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^{n}}\):
\[
6^{-7} = \frac{1}{6^{7}}
\]
Thus, the final answer in exponential form with a positive exponent is:
\[
\frac{1}{6^{7}}
\]