Question
Tyrone claims that the first step to simplify this expression is to raise the numerator and denominator to the third power. Alisha claims that the first step to simplify is to apply the quotient of powers. Who is correct? Explain.
(StartFraction m squared n Superscript negative 3 Baseline Over m Superscript 6 Baseline n Superscript negative 1 Baseline EndFraction) cubed
(StartFraction m squared n Superscript negative 3 Baseline Over m Superscript 6 Baseline n Superscript negative 1 Baseline EndFraction) cubed
Answers
GPT-4o mini
To analyze the expression \( \left( \frac{m^2 n^{-3}}{m^6 n^{-1}} \right)^3 \), let's break down the claims made by Tyrone and Alisha.
### Tyrone's Claim:
Tyrone suggests that the first step is to raise the numerator and denominator to the third power, implying that we should compute the cube of the entire fraction directly:
\[
\left( \frac{m^2 n^{-3}}{m^6 n^{-1}} \right)^3 = \frac{(m^2 n^{-3})^3}{(m^6 n^{-1})^3}
\]
This method is valid, as we can apply the exponent to both the numerator and the denominator. However, it may not be the most efficient first step for simplification.
### Alisha's Claim:
Alisha claims that the first step should be to apply the quotient of powers. This involves simplifying the expression inside the parentheses before applying the cube. So we can apply the quotient of powers property, which states \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{m^2 n^{-3}}{m^6 n^{-1}} = \frac{m^{2-6} n^{-3 - (-1)}}{1} = \frac{m^{-4} n^{-2}}{1} = m^{-4} n^{-2}
\]
Now, we can write this simplified form as:
\[
(m^{-4} n^{-2})^3
\]
Next, we can raise the simplified expression to the third power:
\[
m^{-4 \cdot 3} n^{-2 \cdot 3} = m^{-12} n^{-6}
\]
### Conclusion:
Both Tyrone and Alisha are technically correct in their claims about the process. However, Alisha's approach of simplifying the quotient first before applying the cube is usually more efficient and leads directly to the simplified result.
Thus, **Alisha's method** is generally preferred for simplification. It’s important to apply simplification techniques that reduce complexity before performing operations that can result in larger expressions.
### Tyrone's Claim:
Tyrone suggests that the first step is to raise the numerator and denominator to the third power, implying that we should compute the cube of the entire fraction directly:
\[
\left( \frac{m^2 n^{-3}}{m^6 n^{-1}} \right)^3 = \frac{(m^2 n^{-3})^3}{(m^6 n^{-1})^3}
\]
This method is valid, as we can apply the exponent to both the numerator and the denominator. However, it may not be the most efficient first step for simplification.
### Alisha's Claim:
Alisha claims that the first step should be to apply the quotient of powers. This involves simplifying the expression inside the parentheses before applying the cube. So we can apply the quotient of powers property, which states \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{m^2 n^{-3}}{m^6 n^{-1}} = \frac{m^{2-6} n^{-3 - (-1)}}{1} = \frac{m^{-4} n^{-2}}{1} = m^{-4} n^{-2}
\]
Now, we can write this simplified form as:
\[
(m^{-4} n^{-2})^3
\]
Next, we can raise the simplified expression to the third power:
\[
m^{-4 \cdot 3} n^{-2 \cdot 3} = m^{-12} n^{-6}
\]
### Conclusion:
Both Tyrone and Alisha are technically correct in their claims about the process. However, Alisha's approach of simplifying the quotient first before applying the cube is usually more efficient and leads directly to the simplified result.
Thus, **Alisha's method** is generally preferred for simplification. It’s important to apply simplification techniques that reduce complexity before performing operations that can result in larger expressions.