Square Roots and Cube Roots Test Answer Key with Scoring – Grade 8

Question 1 Evaluate the roots (3 points)

1. 14² = 196, so 14.
2. (−7)³ = −343, so −7.
3. 79 squared is 4981.

(a) 14 (b) −7 (c) 79

Scoring: 1 point per part.

Question 2 Solve for x (3 points)

1. 12² = 144, so x = ±12.
2. (−6)³ = −216, so x = −6.
3. 0.7² = 0.49, so x = ±0.7.

(a) ±12 (b) −6 (c) ±0.7

Scoring: 1 point per part; in (a) and (c) give 1 point only for both signs.

Question 3 Between integers (2 points)

1. 8² = 64 and 9² = 81, so √75 is between 8 and 9.
2. 4³ = 64 and 5³ = 125, so ∛75 is between 4 and 5.

(a) 8 and 9 (b) 4 and 5

Scoring: 1 point per part.

Question 4 Storage bin (3 points)

1. Edge = ∛512 = 8 in, since 8³ = 512.
2. One face: 8² = 64 in².
3. Six faces: 6 · 64 = 384 in².

(a) 8 in (b) 64 in² (c) 384 in²

Scoring: 1 point per part.

Question 5 Two-step equations (3 points)

1. Add 20: 3x² = 147. Divide by 3: x² = 49. So x = ±7.
2. Subtract 6: 2x³ = 128. Divide by 2: x³ = 64, so x = 4.

(a) ±7 (b) 4

Scoring: (a) 1 point for x² = 49, 1 point for both solutions. (b) 1 point.

Question 6 Square deck (3 points)

1. 11² = 121 and 12² = 144, so the side is between 11 and 12 ft.
2. 11.6² = 134.56 and 11.7² = 136.89. Since 135 is closer to 134.56, the side is about 11.6 ft.
3. No. 135 is not a perfect square, so √135 is irrational.

(a) 11 and 12 (b) 11.6 ft (c) No, √135 is irrational

Scoring: 1 point per part.

Question 7 Check Dan’s claim (2 points)

(−5)² = 25, not −25. The square of any real number is zero or positive, so x² = −25 has no real solution.

Dan is wrong: there is no real solution.

Scoring: 1 point for checking (−5)² = 25, 1 point for the explanation about squares.

Question 8 Compare (1 point)

7.1² = 50.41, which is more than 50, so √50 < 7.1.

√50 < 7.1

Scoring: 1 point for the correct comparison with 7.1² = 50.41.