Question
implify this expression using the Quotient Rule of Integer Exponents: 1253÷753 .(1 point)
Responses
553
5 superscript 53 baseline
12106
12 superscript 106 baseline
1253÷753
12 superscript 53 baseline divided by 7 superscript 53 baseline
5106
Responses
553
5 superscript 53 baseline
12106
12 superscript 106 baseline
1253÷753
12 superscript 53 baseline divided by 7 superscript 53 baseline
5106
Answers
Answered by
GPT-4o mini
To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, which states that \( \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \), we can first rewrite the expression:
\[
125^3 \div 75^3 = \frac{125^3}{75^3} = \left(\frac{125}{75}\right)^3
\]
Now, we can simplify \( \frac{125}{75} \):
1. Divide both the numerator and the denominator by 25:
\[
\frac{125 \div 25}{75 \div 25} = \frac{5}{3}
\]
Now we substitute this back into our expression:
\[
\left(\frac{125}{75}\right)^3 = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}
\]
Thus, the simplified expression is \(\frac{125}{27}\). However, looking at your multiple-choice options, it appears they are formatted in a different way. If we need to present the answer with exponents:
\[
\frac{5^3}{3^3} = 5^3 \cdot 3^{-3}
\]
However, it seems there are no options that directly match this simplification of 1253÷753.
If we were to express this as an answer in a similar form provided in the options, the cleaned simplified answer would be most consistent with \( 5^{3} \cdot 3^{-3} \), or equivalently expressed possibly in other formats, however, since there are no directly matching responses here, based on your options and typical convention we could say that the first response is the nearest approach we need, as we haven't gotten an exact match here.
In conclusion, based on how the choices are stacked:
- \( 12^{53} \) doesn’t relate,
- \( 5^{53} \) doesn’t relate,
- \( 12^{53}/7^{53} \) doesn’t relate.
It’s worthwhile to be clear with how the options might fit your lesson’s structure on integer exponents.
In any case, if we assume there was an error or misalignment in formatting or a transcription issue, please check with your source material to ensure we are interpreting the elements correctly.
\[
125^3 \div 75^3 = \frac{125^3}{75^3} = \left(\frac{125}{75}\right)^3
\]
Now, we can simplify \( \frac{125}{75} \):
1. Divide both the numerator and the denominator by 25:
\[
\frac{125 \div 25}{75 \div 25} = \frac{5}{3}
\]
Now we substitute this back into our expression:
\[
\left(\frac{125}{75}\right)^3 = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}
\]
Thus, the simplified expression is \(\frac{125}{27}\). However, looking at your multiple-choice options, it appears they are formatted in a different way. If we need to present the answer with exponents:
\[
\frac{5^3}{3^3} = 5^3 \cdot 3^{-3}
\]
However, it seems there are no options that directly match this simplification of 1253÷753.
If we were to express this as an answer in a similar form provided in the options, the cleaned simplified answer would be most consistent with \( 5^{3} \cdot 3^{-3} \), or equivalently expressed possibly in other formats, however, since there are no directly matching responses here, based on your options and typical convention we could say that the first response is the nearest approach we need, as we haven't gotten an exact match here.
In conclusion, based on how the choices are stacked:
- \( 12^{53} \) doesn’t relate,
- \( 5^{53} \) doesn’t relate,
- \( 12^{53}/7^{53} \) doesn’t relate.
It’s worthwhile to be clear with how the options might fit your lesson’s structure on integer exponents.
In any case, if we assume there was an error or misalignment in formatting or a transcription issue, please check with your source material to ensure we are interpreting the elements correctly.
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