8th Grade Square Roots and Cube Roots Test Solutions and Scoring (PDF)

8th Grade math test solutions: Square Roots and Cube Roots – Maths PDF

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This square cube roots test answers for 8th Grade covers square roots and cube roots (Common Core 8.EE.A.2), with a free PDF to print.

Use this answer key to score the square roots and cube roots test out of 20 points. Each solution shows the full work and a scoring guide, so partial credit can be given fairly when a student shows correct steps but makes an arithmetic slip.

Grade 8 · Common Core State Standards: 8.EE.A.2

Square cube roots test answers: worked solutions

Question 1 – Evaluate roots [2 pts]

Scoring: 1 point for each correct value.

Question 2 – Roots of fractions [2 pts]

\(\frac{3}{7}\)
\(\frac{1}{4}\)

Scoring: 1 point for each correct value.

Question 3 – Solve [3 pts]

\(x = 9\) or \(x = -9\)
\(x = 5\)
No, \(-9\) also works because \((-9)^{2} = 81\).

Scoring: 1 point for each part.

Question 4 – Solve with a coefficient [2 pts]

Divide by 5: \(x^{2} = 16\), so \(x = 4\) or \(x = -4\).

Scoring: 1 point for \(x^{2} = 16\). 1 point for both solutions.

Question 5 – Square plot [3 pts]

Side = 20 m.

Side \(= \sqrt{400} = 20\) m.

Fence \(= 4 \cdot 20 = 80\) m.

Scoring: 1 point for the formula, 1 point for 20 m, 1 point for 80 m.

Question 6 – Cube tank [3 pts]

Edge = 4 ft.

Edge \(= \sqrt[3]{64} = 4\) ft.

Base area \(= 4^{2} = 16\) square feet.

Scoring: 1 point for the setup, 1 point for 4 ft, 1 point for 16 square feet.

Question 7 – Estimate [3 pts]

\(7^{2} = 49\), \(8^{2} = 64\): between 7 and 8.
\(7.7^{2} = 59.29\), \(7.8^{2} = 60.84\): 60 is closer to 59.29 (difference 0.71) than 60.84 (difference 0.84), so about 7.7.
No, 60 is not a perfect square, so the root is irrational.

Scoring: 1 point per part.

Question 8 – Two solutions [2 pts]

Both 3 and -3 square to 9 because a negative times a negative is positive. Only 2 cubes to 8, and \((-2)^{3} = -8\).

Scoring: 1 point for the square case with 3 and -3. 1 point for the cube case.

Total score

Total

20 points

Key facts behind this square cube roots test answers

Key facts

\(\sqrt{81} = 9\) because \(9^{2} = 81\); \(\sqrt[3]{125} = 5\) because \(5^{3} = 125\).
\(x^{2} = p\) with \(p > 0\) has two solutions: \(x = \pm\sqrt{p}\).
\(x^{3} = p\) has one real solution: \(x = \sqrt[3]{p}\).
Know the perfect squares up to \(15^{2}\) and the perfect cubes up to \(5^{3}\).
Roots of non-perfect squares, like \(\sqrt{20}\), are irrational and can be estimated.

How to use this square cube roots test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Grade 8 chapters: Grade 8 math.

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