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Difficulty: MediumPositive and Negative Number Properties

For real numbers pp, qq, and rr, it is given that p<0<q<rp < 0 < q < r. If p3qr=−288p^3 q r = -288 and qrp2=9\frac{q r}{p^2} = 9, what is the value of p+qrp + q r?

Answer: 34

Answer

The value of p+qrp + q r is 34.
From qrp2=9\frac{qr}{p^2} = 9, we get qr=9p2qr = 9p^2. Substituting this into p3qr=−288p^3 qr = -288 yields 9p5=−2889p^5 = -288, so p5=−32p^5 = -32. Taking the fifth root gives p=−2p = -2, which satisfies p<0p < 0. Then qr=9(−2)2=36qr = 9(-2)^2 = 36. Finally, p+qr=−2+36=34p + qr = -2 + 36 = 34.

Step-by-Step Solution

1
Relate qrqr to pp using the given quotient equality
qr=9p2qr = 9p^2
Multiplying both sides of qrp2=9\frac{qr}{p^2} = 9 by p2p^2 isolates qrqr.
2
Substitute qr=9p2qr = 9p^2 into the product equation
p3(9p2)=−288  ⟹  9p5=−288  ⟹  p5=−32p^3 (9p^2) = -288 \implies 9p^5 = -288 \implies p^5 = -32
Replacing qrqr with 9p29p^2 produces a single-variable polynomial in pp.
3
Solve for pp enforcing the sign constraint p<0p < 0
p=−2p = -2
Taking the fifth root of −32-32 yields −2-2, which satisfies p<0p < 0.
4
Calculate the value of qrqr
qr=9(−2)2=36qr = 9(-2)^2 = 36
Squaring a negative number yields a positive value: (−2)2=4(-2)^2 = 4, so 9×4=369 \times 4 = 36.
5
Evaluate the expression p+qrp + qr
p+qr=−2+36=34p + qr = -2 + 36 = 34
Adding p=−2p = -2 and qr=36qr = 36 results in 3434.

Key Concept

Positive and Negative Number Properties
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Topics

Positive and Negative Number Properties

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