Question
Simplify 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
GPT-4 Turbo
To simplify the expression using the Quotient Rule of Integer Exponents, we'll assume that you want to express \( 125^3 ÷ 75^3 \) in its simplest form. First, we need to recognize that \( 125 \) and \( 75 \) are not like bases, so we cannot directly apply the quotient rule. However, both can be expressed in terms of the base 5:
\( 125 = 5^3 \) (because 5 * 5 * 5 = 125)
\( 75 = 5^2 \times 3 \) (because 5 * 5 * 3 = 75)
Now, let's express the original problem with these new bases:
\( 125^3 ÷ 75^3 \)
\( = (5^3)^3 ÷ (5^2 \times 3)^3 \)
According to the power of a power rule, when we raise a power to another power, we multiply the exponents:
\( = 5^{3 \times 3} ÷ (5^{2 \times 3} \times 3^3) \)
\( = 5^9 ÷ (5^6 \times 27) \)
Since \( 3^3 = 27 \), we now have:
\( = \frac{5^9}{5^6 \times 27} \)
We can now apply the quotient rule of integer exponents to the \( 5^9 ÷ 5^6 \) part. The rule states \( a^m ÷ a^n = a^{m-n} \), where a ≠ 0:
\( = \frac{5^9}{5^6} \times \frac{1}{27} \)
\( = 5^{9 - 6} \times \frac{1}{27} \)
\( = 5^3 \times \frac{1}{27} \)
Since \( 5^3 = 125 \) and we have already determined that \( 3^3 = 27 \), our final simplified expression is:
\( \frac{5^3}{27} = \frac{125}{27} \)
This expression doesn't simplify nicely to an integer power, so we leave it as it is. There is no need for further simplification without more specific instructions, such as rounding or converting to a decimal. Therefore, none of the options provided (5^53, 12^106, (12^53 ÷ 7^53), 5^106) are correct representations of the initial expression when it is simplified.
\( 125 = 5^3 \) (because 5 * 5 * 5 = 125)
\( 75 = 5^2 \times 3 \) (because 5 * 5 * 3 = 75)
Now, let's express the original problem with these new bases:
\( 125^3 ÷ 75^3 \)
\( = (5^3)^3 ÷ (5^2 \times 3)^3 \)
According to the power of a power rule, when we raise a power to another power, we multiply the exponents:
\( = 5^{3 \times 3} ÷ (5^{2 \times 3} \times 3^3) \)
\( = 5^9 ÷ (5^6 \times 27) \)
Since \( 3^3 = 27 \), we now have:
\( = \frac{5^9}{5^6 \times 27} \)
We can now apply the quotient rule of integer exponents to the \( 5^9 ÷ 5^6 \) part. The rule states \( a^m ÷ a^n = a^{m-n} \), where a ≠ 0:
\( = \frac{5^9}{5^6} \times \frac{1}{27} \)
\( = 5^{9 - 6} \times \frac{1}{27} \)
\( = 5^3 \times \frac{1}{27} \)
Since \( 5^3 = 125 \) and we have already determined that \( 3^3 = 27 \), our final simplified expression is:
\( \frac{5^3}{27} = \frac{125}{27} \)
This expression doesn't simplify nicely to an integer power, so we leave it as it is. There is no need for further simplification without more specific instructions, such as rounding or converting to a decimal. Therefore, none of the options provided (5^53, 12^106, (12^53 ÷ 7^53), 5^106) are correct representations of the initial expression when it is simplified.