Question
Which of the following is the equivalent expression to (13^−7)^0 ⋅ 7^−11/7^−9 , generated by applying the properties of exponents?
Answers
GPT-4o mini
To simplify the expression \((13^{-7})^0 \cdot \frac{7^{-11}}{7^{-9}}\), we can apply the properties of exponents step by step.
1. **Applying the Zero Exponent Rule**: Any non-zero number raised to the power of 0 is equal to 1. Therefore, we simplify \((13^{-7})^0\) to:
\[
(13^{-7})^0 = 1
\]
2. **Simplifying the Fraction**: Next, we'll simplify the fraction \(\frac{7^{-11}}{7^{-9}}\) using the quotient rule of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{7^{-11}}{7^{-9}} = 7^{-11 - (-9)} = 7^{-11 + 9} = 7^{-2}
\]
3. **Combining the Results**: Now, substitute back into the expression:
\[
1 \cdot 7^{-2} = 7^{-2}
\]
Thus, the equivalent expression is:
\[
\boxed{7^{-2}}
\]
1. **Applying the Zero Exponent Rule**: Any non-zero number raised to the power of 0 is equal to 1. Therefore, we simplify \((13^{-7})^0\) to:
\[
(13^{-7})^0 = 1
\]
2. **Simplifying the Fraction**: Next, we'll simplify the fraction \(\frac{7^{-11}}{7^{-9}}\) using the quotient rule of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{7^{-11}}{7^{-9}} = 7^{-11 - (-9)} = 7^{-11 + 9} = 7^{-2}
\]
3. **Combining the Results**: Now, substitute back into the expression:
\[
1 \cdot 7^{-2} = 7^{-2}
\]
Thus, the equivalent expression is:
\[
\boxed{7^{-2}}
\]