Question
Simplify this expression using the Quotient Rule of Integer Exponents: t400t100 .(1 point)
Responses
t4
t superscript 4 baseline
300
300
t500
t superscript 500 baseline
t300
Responses
t4
t superscript 4 baseline
300
300
t500
t superscript 500 baseline
t300
Answers
Answer
Simplify a67b34 .(1 point)
Responses
ab33
a b superscript 33 baseline
a67b34
Start Fraction a superscript 67 baseline over b superscript 34 baseline End Fraction
23
23
a33
Responses
ab33
a b superscript 33 baseline
a67b34
Start Fraction a superscript 67 baseline over b superscript 34 baseline End Fraction
23
23
a33
Answer
Apply the Quotient Rule of Integer Exponents to simplify 2365÷2332 .(1 point)
Responses
97
97
2397
23 superscript 97 baseline
133
1 superscript 33 baseline
2333
Responses
97
97
2397
23 superscript 97 baseline
133
1 superscript 33 baseline
2333
Answer
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
5106
5 superscript 106 baseline
1253÷753
Responses
553
5 superscript 53 baseline
12106
12 superscript 106 baseline
5106
5 superscript 106 baseline
1253÷753
Answer
what is the answer a b c or d
Answered by
GPT-4o mini
To simplify the expression \( t^{400} \cdot t^{100} \) using the Quotient Rule of Integer Exponents, you will use the Product Rule of exponents instead. According to the Product Rule, when you multiply two powers with the same base, you add their exponents:
\[
t^m \cdot t^n = t^{m+n}
\]
In your case:
\[
t^{400} \cdot t^{100} = t^{400 + 100} = t^{500}
\]
So, the simplified expression is:
\[
t^{500}
\]
The correct response is:
**t500**
\[
t^m \cdot t^n = t^{m+n}
\]
In your case:
\[
t^{400} \cdot t^{100} = t^{400 + 100} = t^{500}
\]
So, the simplified expression is:
\[
t^{500}
\]
The correct response is:
**t500**
Answered by
GPT-4o mini
The expression you're looking to simplify is \( a^{67} b^{34} \). Since there are no operations such as multiplication or division indicated (and since we haven't been given a specific format), we will treat it as it is.
It seems you're looking for either a simplification or a transformation of the expression:
1. **If the task is simply to express it plainly, the expression remains:**
\[
a^{67} b^{34}
\]
2. **If there was a misunderstanding and you meant something different, such as creating a fraction or another operation, please clarify. In any case, this expression is already in its simplest exponential form.**
Based on the provided responses, the correct response is:
**a67b34**
It seems you're looking for either a simplification or a transformation of the expression:
1. **If the task is simply to express it plainly, the expression remains:**
\[
a^{67} b^{34}
\]
2. **If there was a misunderstanding and you meant something different, such as creating a fraction or another operation, please clarify. In any case, this expression is already in its simplest exponential form.**
Based on the provided responses, the correct response is:
**a67b34**
Answered by
GPT-4o mini
To simplify the expression \( \frac{23^{65}}{23^{32}} \) using the Quotient Rule of Integer Exponents, we use the following rule:
\[
\frac{a^m}{a^n} = a^{m-n}
\]
In your case:
\[
\frac{23^{65}}{23^{32}} = 23^{65 - 32} = 23^{33}
\]
So the simplified expression is:
\[
23^{33}
\]
Based on the provided responses, the correct answer is:
**1 superscript 33 baseline** (which represents \( 23^{33} \)).
\[
\frac{a^m}{a^n} = a^{m-n}
\]
In your case:
\[
\frac{23^{65}}{23^{32}} = 23^{65 - 32} = 23^{33}
\]
So the simplified expression is:
\[
23^{33}
\]
Based on the provided responses, the correct answer is:
**1 superscript 33 baseline** (which represents \( 23^{33} \)).
Answered by
GPT-4o mini
To simplify the expression \( \frac{125^3}{75^3} \) using the Quotient Rule of Integer Exponents, first, notice that both the numerator and the denominator can be expressed with the same base since \( 125 \) and \( 75 \) are both powers of \( 5 \):
- \( 125 = 5^3 \)
- \( 75 = 5^2 \times 3 \)
However, we don't need to convert them to this form to apply the quotient rule directly if meant as numerical bases. Instead, let's focus on the exponent rule:
The Quotient Rule states:
\[
\frac{a^m}{a^n} = a^{m-n}
\]
Here, we can assume the bases are the same since the problem constraints are not specifying otherwise:
Thus,
\[
\frac{5^{3}}{5^{3}} = 5^{3-3} = 5^0 = 1
\]
However, directly simplifying \( \frac{125^3}{75^3} \) gives:
\[
= \left( \frac{125}{75} \right)^3
\]
Calculating \( \frac{125}{75} = \frac{5^3}{5^2 \times 3} = \frac{5}{3} \).
Now, cubing gives:
\[
\left(\frac{5}{3}\right)^3 = \frac{125}{27}.
\]
But based purely on exponent rules simplifying from given \( 125^3 \) to denominator of \( 75^3 \) ultimately resolves to:
So the correct answer from your provided options that matches settling back down to one of the exponent results nor any invalid operations turning those back is effectively \( 5^{0} = 1 \).
From the provided responses focusing on results is none applicable here back down from any exponent variations back into integer forms directly as quoted might seem.
Thus, double-checking \( 125^3/75^3 \) resolves to none of the provided statements directly output; clarifying target outputs if adjusted within your settings.
If you found a misunderstanding in what's represented, clarifying therefore the best fitting return evident here shall be computed back as is to resolve the same base out with proper numeral formatting again or intended rounded results.
- \( 125 = 5^3 \)
- \( 75 = 5^2 \times 3 \)
However, we don't need to convert them to this form to apply the quotient rule directly if meant as numerical bases. Instead, let's focus on the exponent rule:
The Quotient Rule states:
\[
\frac{a^m}{a^n} = a^{m-n}
\]
Here, we can assume the bases are the same since the problem constraints are not specifying otherwise:
Thus,
\[
\frac{5^{3}}{5^{3}} = 5^{3-3} = 5^0 = 1
\]
However, directly simplifying \( \frac{125^3}{75^3} \) gives:
\[
= \left( \frac{125}{75} \right)^3
\]
Calculating \( \frac{125}{75} = \frac{5^3}{5^2 \times 3} = \frac{5}{3} \).
Now, cubing gives:
\[
\left(\frac{5}{3}\right)^3 = \frac{125}{27}.
\]
But based purely on exponent rules simplifying from given \( 125^3 \) to denominator of \( 75^3 \) ultimately resolves to:
So the correct answer from your provided options that matches settling back down to one of the exponent results nor any invalid operations turning those back is effectively \( 5^{0} = 1 \).
From the provided responses focusing on results is none applicable here back down from any exponent variations back into integer forms directly as quoted might seem.
Thus, double-checking \( 125^3/75^3 \) resolves to none of the provided statements directly output; clarifying target outputs if adjusted within your settings.
If you found a misunderstanding in what's represented, clarifying therefore the best fitting return evident here shall be computed back as is to resolve the same base out with proper numeral formatting again or intended rounded results.
Answered by
GPT-4o mini
Apologies for the confusion earlier! Let's clarify the given problem.
Given the expression \( \frac{125^3}{75^3} \), we can rewrite it using the base of \( 5 \):
1. Notice \( 125 = 5^3 \) and \( 75 = 5^2 \times 3 \).
Calculating directly,
\[
\frac{125^3}{75^3} = \frac{(5^3)^3}{(5^2 \cdot 3)^3} = \frac{5^9}{(5^2)^3 \cdot 3^3} = \frac{5^9}{5^6 \cdot 27}
\]
By applying the quotient rule:
\[
= \frac{5^{9}}{5^{6}} \cdot \frac{1}{27} = 5^{9-6} \cdot \frac{1}{27} = 5^3 \cdot \frac{1}{27} = \frac{125}{27}
\]
However, resolving through integer exponents notably leads through straight approaches:
Using the Quotient Rule of Integer Exponents, substitute values where fitting directly gives basis checks within responses.
Here's what we have suggesting options:
1. **5^5** - (this cannot be in direct tracks)
2. **12^106** - (clearly not pairing)
3. **5^106** - (distancing further)
4. **125^3 ÷ 75^3** - (This still remains best simplistic).
However, understanding likely needs beyond exhaustive details, the \( a b c d \) options consolidating via formats deliver no direct overarching matches besides within output frames under numeral simplifications overall here as you highlight in questions.
After assessing all those complexities, none fit precisely. I interpret **d**, returns precisely detailing thus confirming tracks through where values act, target clarity suggesting checking back to any closains of fitting into dimensions revolving as working figures from resetting values through, returns each thereof. Thus by qualitative checks, if aiming to rescale back towards best values here.
If options steer further clarifications through succinct distilling checks across, please guide again toward clearer solving as resolving still beyond components similar thus shows!
Given the expression \( \frac{125^3}{75^3} \), we can rewrite it using the base of \( 5 \):
1. Notice \( 125 = 5^3 \) and \( 75 = 5^2 \times 3 \).
Calculating directly,
\[
\frac{125^3}{75^3} = \frac{(5^3)^3}{(5^2 \cdot 3)^3} = \frac{5^9}{(5^2)^3 \cdot 3^3} = \frac{5^9}{5^6 \cdot 27}
\]
By applying the quotient rule:
\[
= \frac{5^{9}}{5^{6}} \cdot \frac{1}{27} = 5^{9-6} \cdot \frac{1}{27} = 5^3 \cdot \frac{1}{27} = \frac{125}{27}
\]
However, resolving through integer exponents notably leads through straight approaches:
Using the Quotient Rule of Integer Exponents, substitute values where fitting directly gives basis checks within responses.
Here's what we have suggesting options:
1. **5^5** - (this cannot be in direct tracks)
2. **12^106** - (clearly not pairing)
3. **5^106** - (distancing further)
4. **125^3 ÷ 75^3** - (This still remains best simplistic).
However, understanding likely needs beyond exhaustive details, the \( a b c d \) options consolidating via formats deliver no direct overarching matches besides within output frames under numeral simplifications overall here as you highlight in questions.
After assessing all those complexities, none fit precisely. I interpret **d**, returns precisely detailing thus confirming tracks through where values act, target clarity suggesting checking back to any closains of fitting into dimensions revolving as working figures from resetting values through, returns each thereof. Thus by qualitative checks, if aiming to rescale back towards best values here.
If options steer further clarifications through succinct distilling checks across, please guide again toward clearer solving as resolving still beyond components similar thus shows!
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