Asked by lexi

Selena and Drake are evaluating the expression (StartFraction r s Superscript negative 2 Baseline Over r squared s Superscript negative 3 Baseline EndFraction) Superscript negative 1, when r = negative 1 and s = negative 2.


Selena’s Work
Drake’s Work

(StartFraction r s Superscript negative 2 Baseline Over r squared s Superscript negative 3 Baseline EndFraction) Superscript negative 1 Baseline = (r Superscript negative 1 Baseline s) Superscript negative 1 Baseline = StartFraction r Over s EndFraction = StartFraction negative 1 Over negative 2 EndFraction = one-half

(StartFraction (negative 1) (negative 2) Superscript negative 2 Baseline Over (negative 1) squared (negative 2) Superscript negative 3 EndFraction) Superscript negative 1 = (StartFraction (negative 1) (negative 2) cubed Over (negative 1) squared (negative 2) squared EndFraction) Superscript negative 1 = (StartFraction negative 8 Over 4 Endfraction) Superscript negative 1 Baseline = StartFraction 4 Over negative 8 EndFraction = negative one-half


Who is correct and why?
Selena is incorrect because she should have substituted the values for the variables first, and then simplifed.
Selena is correct because she simplified correctly and then evaluated correctly after substituting the values for the variables.
Drake is incorrect because he should have simplified first, before substituting the values for the variables.
Drake is correct because he substituted the values for the variables first, and simplified correctly.

Geoffrey is evaluating the expression StartFraction (negative 3) cubed (2 Superscript 6 Baseline) Over (Negative 3) Superscript 5 Baseline (2 squared) EndFraction as shown below.
StartFraction (negative 3) cubed (2 Superscript 6 Baseline) Over (Negative 3) Superscript 5 Baseline (2 squared) EndFraction = StartFraction (2) Superscript a Baseline Over (negative 3) Superscript b Baseline EndFraction = StartFraction c Over d EndFraction

What are the values of a, b, c, and d?
a = 4, b = 2, c = 16, d = 9
a = 4, b = negative 2, c = 16, d = 9
a = 8, b = 8, c = 256, d = 6,561
a = 8, b = 8, c = 256, d = negative 6,561

Which is the value of this expression when j = negative 2 and k = negative 1?

(StartFraction j k Superscript negative 2 Baseline Over j Superscript negative 1 Baseline k Superscript negative 3 Baseline EndFraction) cubed
–64
Negative one-half
One-half
64

Which is the value of this expression when x = negative 2 and y = negative 3?

10 x cubed y squared
Negative 720
Negative 360
360
720

Which is the value of this expression when p = negative 2 and q = negative 1?

((p squared) (q Superscript negative 3 Baseline)) Superscript negative 2 Baseline times ((p Superscript negative 3 Baseline) (q superscript 5 Baseline)) Superscript negative 2
–4
Negative StartFraction 1 Over 16 EndFraction
StartFraction 1 Over 16 EndFraction
4

Rena used the steps below to evaluate the expression (StartFraction (x Superscript negative 3 Baseline) (y Superscript negative 2 Baseline) Over 2 (x Superscript 4 Baseline) (y superscript negative 4 Baseline) EndFraction) Superscript negative 3, when x = negative 1 and y = 2.

Step 1:
Substitute x = negative 1 and y = 2 into the expression.
(StartFraction (negative 1) Superscript negative 3 Baseline (2) Superscript negative 2 Baseline Over 2 (negative 1) Superscript 4 Baseline (2) superscript negative 4 Baseline) EndFraction) Superscript negative 3
Step 2:
Simplify the parentheses.
(StartFraction (2) Superscript 4 Baseline Over 2 (negative 1) Superscript 4 Baseline (negative 1) cubed (2) squared EndFraction) Superscript negative 3 Baseline = (StartFraction (2) squared Over 2 (negative 1) Superscript 7 Baseline EndFraction) Superscript negative 3
Step 3:
Evaluate the power to a power.
StartFraction (2) Superscript negative 6 Baseline Over 2 Superscript negative 3 Baseline (negative 1) Superscript negative 21 baseline EndFraction
Step 4:
Use reciprocals and find the value.
StartFraction 1 Over 2 cubed (2) Superscript 6 Baseline (negative 1) Superscript 21 Baseline EndFraction = StartFraction 1 Over 8 times 64 times (negative 1) EndFraction = Negative StartFraction 1 Over 512 EndFraction

In which step did Rena make the first error?
Step 1
Step 2
Step 3
Step 4

Which is the simplified form of the expression StartFraction (6 Superscript negative 4 Baseline) Superscript negative 9 Baseline Over 6 Superscript 6 Baseline EndFraction?
StartFraction 1 Over 6 Superscript 42 Baseline EndFraction
StartFraction 1 Over 6 Superscript 19 Baseline EndFraction
6 Superscript 6
6 Superscript 30

All Answers 1

Answered by GPT-5 mini AI
1) Selena vs. Drake:
- Selena is correct because she simplified correctly and then evaluated correctly after substituting the values for the variables.
- Reason: ((r s^{-2})/(r^2 s^{-3}))^{-1} = (r^{-1} s)^{-1} = r/s. With r = −1, s = −2 gives (−1)/(−2) = 1/2.

2) Geoffrey: a = 4, b = 2, c = 16, d = 9.
- (-3)^3 2^6 / [(-3)^5 2^2] = (-3)^{3-5} 2^{6-2} = (-3)^{-2} 2^4 = 2^4 / (-3)^2 = 16/9.

3) Value when j = −2, k = −1 of ((j k^{-2})/(j^{-1} k^{-3}))^3:
- Answer: −64.
- Simplify: (j^2 k)^3 = j^6 k^3; (-2)^6(−1)^3 = 64·(−1) = −64.

4) Value when x = −2, y = −3 of 10 x^3 y^2:
- Answer: −720.
- 10(−2)^3(−3)^2 = 10(−8)(9) = −720.

5) Value when p = −2, q = −1 of ((p^2 q^{-3})^{-2} (p^{-3} q^5)^{-2}):
- Answer: 4.
- Simplify: p^2 q^{-4}; with p = −2, q = −1 gives 4/1 = 4.

6) Rena’s first error:
- Step 2.
- She made incorrect algebraic simplifications when simplifying inside the parentheses.

7) Simplified form of (6^{-4})^{-9} / 6^6:
- Answer: 6^{30}.
- (6^{-4})^{-9} = 6^{36}; 6^{36}/6^6 = 6^{30}.