Question
Explain how the Quotient of Powers Property was used to simplify this expression.
5 to the fourth power, over 25 = 52
a By simplifying 25 to 52 to make both powers base five and subtracting the exponents
b By simplifying 25 to 52 to make both powers base five and adding the exponents
c By finding the quotient of the bases to be one fifth and cancelling common factors
d By finding the quotient of the bases to be one fifth and simplifying the expression
5 to the fourth power, over 25 = 52
a By simplifying 25 to 52 to make both powers base five and subtracting the exponents
b By simplifying 25 to 52 to make both powers base five and adding the exponents
c By finding the quotient of the bases to be one fifth and cancelling common factors
d By finding the quotient of the bases to be one fifth and simplifying the expression
Answers
GPT-4o mini
To analyze how the Quotient of Powers Property was used to simplify the expression \( \frac{5^4}{25} = 5^2 \), let's break down the options given:
1. **Understanding the expression**:
The expression \( \frac{5^4}{25} \) can be rewritten because \( 25 \) is \( 5^2 \). Therefore, we have:
\[
\frac{5^4}{25} = \frac{5^4}{5^2}
\]
2. **Applying the Quotient of Powers Property**:
The Quotient of Powers Property states that when dividing powers with the same base, you subtract the exponents. In this case:
\[
\frac{5^4}{5^2} = 5^{4-2} = 5^2
\]
Now, looking at the choices given:
- **a) By simplifying 25 to 52 to make both powers base five and subtracting the exponents**: This statement is correct because it explains that you rewrite \( 25 \) as \( 5^2 \) and then apply the property correctly by subtracting exponents.
- **b) By simplifying 25 to 52 to make both powers base five and adding the exponents**: This is incorrect because you subtract the exponents, not add them.
- **c) By finding the quotient of the bases to be one fifth and cancelling common factors**: This is not how the Quotient of Powers Property is applied here, so it does not explain the simplification accurately.
- **d) By finding the quotient of the bases to be one fifth and simplifying the expression**: This implies a method not used here, as we didn't directly find the quotient of different bases.
From this analysis, the best answer is **a)** because it correctly explains the process of simplification as applied in this expression.
1. **Understanding the expression**:
The expression \( \frac{5^4}{25} \) can be rewritten because \( 25 \) is \( 5^2 \). Therefore, we have:
\[
\frac{5^4}{25} = \frac{5^4}{5^2}
\]
2. **Applying the Quotient of Powers Property**:
The Quotient of Powers Property states that when dividing powers with the same base, you subtract the exponents. In this case:
\[
\frac{5^4}{5^2} = 5^{4-2} = 5^2
\]
Now, looking at the choices given:
- **a) By simplifying 25 to 52 to make both powers base five and subtracting the exponents**: This statement is correct because it explains that you rewrite \( 25 \) as \( 5^2 \) and then apply the property correctly by subtracting exponents.
- **b) By simplifying 25 to 52 to make both powers base five and adding the exponents**: This is incorrect because you subtract the exponents, not add them.
- **c) By finding the quotient of the bases to be one fifth and cancelling common factors**: This is not how the Quotient of Powers Property is applied here, so it does not explain the simplification accurately.
- **d) By finding the quotient of the bases to be one fifth and simplifying the expression**: This implies a method not used here, as we didn't directly find the quotient of different bases.
From this analysis, the best answer is **a)** because it correctly explains the process of simplification as applied in this expression.