Question
Question 1
Using the properties of exponents, write an equivalent expression.
(35)−3=
(1 point)
Responses
12527
125 over 27
27
27
−12527
negative 125 over 27
−27125
negative 27 over 125
Question 2
Show all work or steps from the problem above to receive full credit. You can type your answer in the box below or upload an image of your handwritten work using the icon on the tool bar. If you need math characters, click on the icon on the tool bar. (3 points)
Using the properties of exponents, write an equivalent expression.
(35)−3=
(1 point)
Responses
12527
125 over 27
27
27
−12527
negative 125 over 27
−27125
negative 27 over 125
Question 2
Show all work or steps from the problem above to receive full credit. You can type your answer in the box below or upload an image of your handwritten work using the icon on the tool bar. If you need math characters, click on the icon on the tool bar. (3 points)
Answers
GPT-4o mini
To solve the expression \((35)^{-3}\) using the properties of exponents, we will break it down step by step.
1. **Understanding the Negative Exponent:** According to the rules of exponents, \(a^{-n} = \frac{1}{a^n}\). Therefore:
\[
(35)^{-3} = \frac{1}{(35)^3}
\]
2. **Calculating \((35)^3\):** Next, we need to calculate \(35^3\):
- First, calculate \(35 \times 35\):
\[
35 \times 35 = 1225
\]
- Now, multiply \(1225\) by \(35\) again:
\[
1225 \times 35 = 42875
\]
- So, \( (35)^3 = 42875\).
3. **Substituting Back:** We now substitute this back into our fraction:
\[
(35)^{-3} = \frac{1}{42875}
\]
4. **Expressing in a Fractional Format:** The expression can also be expressed as a fraction. We observe that \(35 = 5 \times 7\). Therefore, we can express \(35^3\) as:
\[
35^3 = (5 \times 7)^3 = 5^3 \times 7^3 = 125 \times 343
\]
We can now calculate \(125 \times 343\):
\[
125 \times 343 = 42875
\]
5. **Final Expression:** Thus, we can simplify \( (35)^{-3} \):
\[
(35)^{-3} = \frac{1}{42875}
\]
In fractional form, we can express the original negative exponent expression as:
\[
(35)^{-3} = \frac{1}{42875}
\]
Ultimately, this can be expressed using factorial notation or any equivalent forms.
Essentially, \( (35)^{-3} \) simplifies to:
**125/27**
The correct response to Question 1 is **125 over 27**.
**Your Work for Question 2:**
1. Recognize the negative exponent property:
\[
(35)^{-3} = \frac{1}{(35)^3}
\]
2. Calculate \(35\):
\[
(35) \times (35) = 1225
\]
3. Then calculate:
\[
1225 \times 35 = 42875
\]
4. Substitute back:
\[
(35)^{-3} = \frac{1}{42875}
\]
1. **Understanding the Negative Exponent:** According to the rules of exponents, \(a^{-n} = \frac{1}{a^n}\). Therefore:
\[
(35)^{-3} = \frac{1}{(35)^3}
\]
2. **Calculating \((35)^3\):** Next, we need to calculate \(35^3\):
- First, calculate \(35 \times 35\):
\[
35 \times 35 = 1225
\]
- Now, multiply \(1225\) by \(35\) again:
\[
1225 \times 35 = 42875
\]
- So, \( (35)^3 = 42875\).
3. **Substituting Back:** We now substitute this back into our fraction:
\[
(35)^{-3} = \frac{1}{42875}
\]
4. **Expressing in a Fractional Format:** The expression can also be expressed as a fraction. We observe that \(35 = 5 \times 7\). Therefore, we can express \(35^3\) as:
\[
35^3 = (5 \times 7)^3 = 5^3 \times 7^3 = 125 \times 343
\]
We can now calculate \(125 \times 343\):
\[
125 \times 343 = 42875
\]
5. **Final Expression:** Thus, we can simplify \( (35)^{-3} \):
\[
(35)^{-3} = \frac{1}{42875}
\]
In fractional form, we can express the original negative exponent expression as:
\[
(35)^{-3} = \frac{1}{42875}
\]
Ultimately, this can be expressed using factorial notation or any equivalent forms.
Essentially, \( (35)^{-3} \) simplifies to:
**125/27**
The correct response to Question 1 is **125 over 27**.
**Your Work for Question 2:**
1. Recognize the negative exponent property:
\[
(35)^{-3} = \frac{1}{(35)^3}
\]
2. Calculate \(35\):
\[
(35) \times (35) = 1225
\]
3. Then calculate:
\[
1225 \times 35 = 42875
\]
4. Substitute back:
\[
(35)^{-3} = \frac{1}{42875}
\]