Grade 8 Algebra: Square Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 1

Perfect squares

1. Evaluate:
$\sqrt{81}$
2. Evaluate:
$\sqrt{625}$
3. Evaluate:
$\sqrt{2401}$
4. Evaluate:
$\sqrt{4096}$
5. Evaluate:
$\sqrt{6561}$
6. Evaluate:
$\sqrt{10000}$
7. Evaluate:
$\sqrt{20736}$
8. Evaluate:
$\sqrt{28561}$

Answer Key

Level 1

1. $9$
2. $25$
3. $49$
4. $64$
5. $81$
6. $100$
7. $144$
8. $169$

Grade 8 Algebra: Square Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 2

Fractions, decimals and negatives

1. Evaluate:
$\sqrt{\tfrac{4}{9}}$
2. Evaluate:
$\sqrt{\tfrac{16}{25}}$
3. Evaluate:
$\sqrt{\tfrac{1}{36}}$
4. Evaluate:
$\sqrt{0.49}$
5. Evaluate:
$\sqrt{0.01}$
6. Evaluate:
$-\sqrt{64}$
7. Evaluate:
$\pm\sqrt{121}$
8. Evaluate:
$\sqrt{1.44}$

Answer Key

Level 2

1. $\tfrac{2}{3}$
2. $\tfrac{4}{5}$
3. $\tfrac{1}{6}$
4. $0.7$
5. $0.1$
6. $-8$
7. $\pm 11$
8. $1.2$

Grade 8 Algebra: Square Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 3

Expressions with square roots

1. Evaluate:
$\sqrt{16}+\sqrt{9}$
2. Evaluate:
$3\sqrt{25}$
3. Evaluate:
$\sqrt{100}-2\sqrt{4}$
4. Evaluate:
$\sqrt{9+16}$
5. Evaluate:
$\sqrt{36}\cdot\sqrt{4}$
6. Evaluate:
$\dfrac{\sqrt{81}}{\sqrt{9}}$
7. Evaluate:
$(\sqrt{7})^2$
8. Evaluate:
$\sqrt{5^2-3^2}$

Answer Key

Level 3

1. $7$
2. $15$
3. $6$
4. $5$
5. $12$
6. $3$
7. $7$
8. $4$

Grade 8 Algebra: Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 1

Perfect cubes

1. Evaluate:
$\sqrt[3]{1}$
2. Evaluate:
$\sqrt[3]{512}$
3. Evaluate:
$\sqrt[3]{19683}$
4. Evaluate:
$\sqrt[3]{262144}$
5. Evaluate:
$\sqrt[3]{1953125}$
6. Evaluate:
$\sqrt[3]{10077696}$
7. Evaluate:
$\sqrt[3]{1000000000}$
8. Evaluate:
$\sqrt[3]{40353607}$

Answer Key

Level 1

1. $1$
2. $8$
3. $27$
4. $64$
5. $125$
6. $216$
7. $1000$
8. $343$

Grade 8 Algebra: Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 2

Negatives and fractions

1. Evaluate:
$\sqrt[3]{-8}$
2. Evaluate:
$\sqrt[3]{-27}$
3. Evaluate:
$\sqrt[3]{-125}$
4. Evaluate:
$\sqrt[3]{\tfrac{1}{8}}$
5. Evaluate:
$\sqrt[3]{\tfrac{8}{27}}$
6. Evaluate:
$\sqrt[3]{0.001}$
7. Evaluate:
$\sqrt[3]{-\tfrac{64}{125}}$
8. Evaluate:
$\sqrt[3]{0.008}$

Answer Key

Level 2

1. $-2$
2. $-3$
3. $-5$
4. $\tfrac{1}{2}$
5. $\tfrac{2}{3}$
6. $0.1$
7. $-\tfrac{4}{5}$
8. $0.2$

Grade 8 Algebra: Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 3

Expressions with cube roots

1. Evaluate:
$\sqrt[3]{8}+\sqrt[3]{27}$
2. Evaluate:
$2\sqrt[3]{64}$
3. Evaluate:
$\sqrt[3]{125}-\sqrt{25}$
4. Evaluate:
$\sqrt[3]{-1}+\sqrt{1}$
5. Evaluate:
$\sqrt[3]{1000}\cdot\sqrt[3]{8}$
6. Evaluate:
$\dfrac{\sqrt[3]{216}}{\sqrt[3]{-27}}$
7. Evaluate:
$(\sqrt[3]{5})^3$
8. Evaluate:
$\sqrt[3]{2^3+19}$

Answer Key

Level 3

1. $5$
2. $8$
3. $0$
4. $0$
5. $20$
6. $-2$
7. $5$
8. $3$

Grade 8 Algebra: Estimating Roots and Irrational Numbers

Name: ______________________Date: ____________Score: ____ / 8

Level 1

Between two whole numbers

1. Name the two whole numbers it lies between:
$\sqrt{5}$
2. Name the two whole numbers it lies between:
$\sqrt{10}$
3. Name the two whole numbers it lies between:
$\sqrt{20}$
4. Name the two whole numbers it lies between:
$\sqrt{30}$
5. Name the two whole numbers it lies between:
$\sqrt{45}$
6. Name the two whole numbers it lies between:
$\sqrt{60}$
7. Name the two whole numbers it lies between:
$\sqrt{75}$
8. Name the two whole numbers it lies between:
$\sqrt{90}$

Answer Key

Level 1

1. $2 \text{ and } 3$
2. $3 \text{ and } 4$
3. $4 \text{ and } 5$
4. $5 \text{ and } 6$
5. $6 \text{ and } 7$
6. $7 \text{ and } 8$
7. $8 \text{ and } 9$
8. $9 \text{ and } 10$

Grade 8 Algebra: Estimating Roots and Irrational Numbers

Name: ______________________Date: ____________Score: ____ / 8

Level 2

Nearest whole number

1. Estimate to the nearest whole number:
$\sqrt{8}$
2. Estimate to the nearest whole number:
$\sqrt{15}$
3. Estimate to the nearest whole number:
$\sqrt{26}$
4. Estimate to the nearest whole number:
$\sqrt{40}$
5. Estimate to the nearest whole number:
$\sqrt{50}$
6. Estimate to the nearest whole number:
$\sqrt{63}$
7. Estimate to the nearest whole number:
$\sqrt{80}$
8. Estimate to the nearest whole number:
$\sqrt{99}$

Answer Key

Level 2

1. $3$
2. $4$
3. $5$
4. $6$
5. $7$
6. $8$
7. $9$
8. $10$

Grade 8 Algebra: Estimating Roots and Irrational Numbers

Name: ______________________Date: ____________Score: ____ / 8

Level 3

Rational or irrational, and ordering

1. Rational or irrational?:
$\sqrt{49}$
2. Rational or irrational?:
$\sqrt{12}$
3. Rational or irrational?:
$\sqrt[3]{27}$
4. Rational or irrational?:
$\pi$
5. Rational or irrational?:
$\sqrt{\tfrac{9}{16}}$
6. Order from least to greatest:
$\sqrt{10},\; 3.5,\; \pi$
7. Order from least to greatest:
$\sqrt{50},\; 7.2,\; \sqrt{48}$
8. Estimate to one decimal place:
$\sqrt{30}$

Answer Key

Level 3

1. $\text{rational}$
2. $\text{irrational}$
3. $\text{rational}$
4. $\text{irrational}$
5. $\text{rational}$
6. $\pi,\; \sqrt{10},\; 3.5$
7. $\sqrt{48},\; \sqrt{50},\; 7.2$
8. $\approx 5.5$

Grade 8 Algebra: Solving x² = p and x³ = p

Name: ______________________Date: ____________Score: ____ / 8

Level 1

Perfect squares and cubes

1. Solve:
$x^2=36$
2. Solve:
$x^2=81$
3. Solve:
$x^3=8$
4. Solve:
$x^3=125$
5. Solve:
$x^2=144$
6. Solve:
$x^3=-27$
7. Solve:
$x^2=1$
8. Solve:
$x^3=1000$

Answer Key

Level 1

1. $x=\pm 6$
2. $x=\pm 9$
3. $x=2$
4. $x=5$
5. $x=\pm 12$
6. $x=-3$
7. $x=\pm 1$
8. $x=10$

Grade 8 Algebra: Solving x² = p and x³ = p

Name: ______________________Date: ____________Score: ____ / 8

Level 2

Rearrange first

1. Solve:
$x^2+5=30$
2. Solve:
$2x^2=50$
3. Solve:
$x^3-4=60$
4. Solve:
$3x^3=24$
5. Solve:
$x^2-7=42$
6. Solve:
$\tfrac{x^2}{2}=32$
7. Solve:
$x^3+10=-54$
8. Solve:
$4x^2-1=99$

Answer Key

Level 2

1. $x=\pm 5$
2. $x=\pm 5$
3. $x=4$
4. $x=2$
5. $x=\pm 7$
6. $x=\pm 8$
7. $x=-4$
8. $x=\pm 5$

Grade 8 Algebra: Solving x² = p and x³ = p

Name: ______________________Date: ____________Score: ____ / 8

Level 3

Fractions and irrational answers

1. Solve:
$x^2=\tfrac{9}{25}$
2. Solve:
$x^3=\tfrac{1}{64}$
3. Solve:
$x^2=7$
4. Solve:
$x^3=10$
5. Solve:
$x^2=0.16$
6. Solve:
$5x^2=15$
7. Solve:
$x^3=-\tfrac{8}{125}$
8. Solve:
$2x^3+3=19$

Answer Key

Level 3

1. $x=\pm\tfrac{3}{5}$
2. $x=\tfrac{1}{4}$
3. $x=\pm\sqrt{7}$
4. $x=\sqrt[3]{10}$
5. $x=\pm 0.4$
6. $x=\pm\sqrt{3}$
7. $x=-\tfrac{2}{5}$
8. $x=2$

Grade 8 Algebra: Word Problems: Square and Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 1

Squares and cubes

1. Solve:
$\text{A}$ $\text{square}$ $\text{rug}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{36}$ $\text{square}$ $\text{feet.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{each}$ $\text{side?}$
2. Solve:
$\text{A}$ $\text{cube}$ $\text{box}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{64}$ $\text{cubic}$ $\text{inches.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{each}$ $\text{edge?}$
3. Solve:
$\text{A}$ $\text{square}$ $\text{garden}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{81}$ $\text{square}$ $\text{meters.}$ $\text{What}$ $\text{is}$ $\text{the}$ $\text{length}$ $\text{of}$ $\text{one}$ $\text{side?}$
4. Solve:
$\text{A}$ $\text{cube}$ $\text{of}$ $\text{ice}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{27}$ $\text{cubic}$ $\text{cm.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{each}$ $\text{edge?}$
5. Solve:
$\text{A}$ $\text{square}$ $\text{tile}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{144}$ $\text{square}$ $\text{cm.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{each}$ $\text{side?}$
6. Solve:
$\text{A}$ $\text{cube}$ $\text{tank}$ $\text{holds}$ $\text{1000}$ $\text{cubic}$ $\text{feet.}$ $\text{How}$ $\text{tall}$ $\text{is}$ $\text{it?}$
7. Solve:
$\text{A}$ $\text{square}$ $\text{photo}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{25}$ $\text{square}$ $\text{inches.}$ $\text{What}$ $\text{is}$ $\text{its}$ $\text{perimeter?}$
8. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{125}$ $\text{cubic}$ $m\text{.}$ $\text{What}$ $\text{is}$ $\text{the}$ $\text{area}$ $\text{of}$ $\text{one}$ $\text{face?}$

Answer Key

Level 1

1. $6\text{ ft}$
2. $4\text{ in}$
3. $9\text{ m}$
4. $3\text{ cm}$
5. $12\text{ cm}$
6. $10\text{ ft}$
7. $20\text{ in}$
8. $25\text{ m}^2$

Grade 8 Algebra: Word Problems: Square and Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 2

Working backwards

1. Solve:
$\text{A}$ $\text{square}$ $\text{park}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{400}$ $\text{square}$ $m\text{.}$ $\text{How}$ $\text{much}$ $\text{fence}$ $\text{goes}$ $\text{around}$ $\text{it?}$
2. Solve:
$\text{A}$ $\text{square}$ $\text{floor}$ $\text{needs}$ $\text{196}$ $\text{square}$ $\text{tiles}$ $\text{of}$ $\text{side}$ $\text{1}$ $\text{ft.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{the}$ $\text{floor?}$
3. Solve:
$\text{Two}$ $\text{equal}$ $\text{cubes}$ $\text{have}$ $\text{a}$ $\text{total}$ $\text{volume}$ $\text{of}$ $\text{54}$ $\text{cubic}$ $\text{cm.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{each}$ $\text{edge?}$
4. Solve:
$\text{A}$ $\text{number}$ $\text{squared}$ $\text{plus}$ $\text{11}$ $\text{equals}$ $\text{60.}$ $\text{What}$ $\text{could}$ $\text{the}$ $\text{number}$ $n\text{}$ $\text{be?}$
5. Solve:
$\text{A}$ $\text{number}$ $\text{cubed}$ $\text{minus}$ $\text{5}$ $\text{equals}$ $\text{3.}$ $\text{What}$ $\text{is}$ $\text{the}$ $\text{number}$ $n\text{?}$
6. Solve:
$\text{A}$ $\text{square}$ $\text{has}$ $\text{the}$ $\text{same}$ $\text{area}$ $\text{as}$ $\text{a}$ $\text{4}$ $m\text{}$ $\text{by}$ $\text{9}$ $m\text{}$ $\text{rectangle.}$ $\text{How}$ $\text{long}$ $\text{is}$ $\text{its}$ $\text{side?}$
7. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{216}$ $\text{cubic}$ $\text{in.}$ $\text{What}$ $\text{is}$ $\text{its}$ $\text{surface}$ $\text{area?}$
8. Solve:
$\text{A}$ $\text{square}$ $\text{poster}$ $\text{has}$ $\text{a}$ $\text{perimeter}$ $\text{of}$ $\text{48}$ $\text{in.}$ $\text{What}$ $\text{is}$ $\text{its}$ $\text{area?}$

Answer Key

Level 2

1. $80\text{ m}$
2. $14\text{ ft}$
3. $3\text{ cm}$
4. $n=\pm 7$
5. $n=2$
6. $6\text{ m}$
7. $216\text{ in}^2$
8. $144\text{ in}^2$

Grade 8 Algebra: Word Problems: Square and Cube Roots

Name: ______________________Date: ____________Score: ____ / 8

Level 3

Estimating and comparing

1. Solve:
$\text{A}$ $\text{square}$ $\text{room}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{50}$ $\text{square}$ $\text{ft.}$ $\text{Between}$ $\text{which}$ $\text{two}$ $\text{whole}$ $\text{numbers}$ $\text{is}$ $\text{its}$ $\text{side}$ $\text{length?}$
2. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{30}$ $\text{cubic}$ $\text{cm.}$ $\text{Between}$ $\text{which}$ $\text{two}$ $\text{whole}$ $\text{numbers}$ $\text{is}$ $\text{its}$ $\text{edge?}$
3. Solve:
$\text{A}$ $\text{square}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{20}$ $\text{square}$ $m\text{.}$ $\text{Estimate}$ $\text{its}$ $\text{side}$ $\text{to}$ $\text{the}$ $\text{nearest}$ $\text{whole}$ $\text{meter.}$
4. Solve:
$\text{One}$ $\text{square}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{49}$ $\text{square}$ $\text{cm}$ $\text{and}$ $\text{a}$ $\text{bigger}$ $\text{square}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{64}$ $\text{square}$ $\text{cm.}$ $\text{How}$ $\text{much}$ $\text{longer}$ $\text{is}$ $\text{a}$ $\text{side}$ $\text{of}$ $\text{the}$ $\text{bigger}$ $\text{square?}$
5. Solve:
$\text{A}$ $\text{ladder}$ $\text{reaches}$ $\text{a}$ $\text{height}$ $\text{whose}$ $\text{square}$ $\text{is}$ $\text{90.}$ $\text{Is}$ $\text{the}$ $\text{height}$ $\text{more}$ $\text{or}$ $\text{less}$ $\text{than}$ $\text{9}$ $m\text{?}$
6. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{a}$ $\text{volume}$ $\text{of}$ $\text{8}$ $\text{cubic}$ $m\text{.}$ $\text{If}$ $\text{each}$ $\text{edge}$ $\text{is}$ $\text{doubled,}$ $\text{what}$ $\text{is}$ $\text{the}$ $\text{new}$ $\text{volume?}$
7. Solve:
$\text{A}$ $\text{square}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{2}$ $\text{square}$ $\text{units.}$ $\text{Is}$ $\text{its}$ $\text{side}$ $\text{rational}$ $\text{or}$ $\text{irrational?}$
8. Solve:
$\text{A}$ $\text{square}$ $\text{field}$ $\text{has}$ $\text{an}$ $\text{area}$ $\text{of}$ $\text{10000}$ $\text{square}$ $m\text{.}$ $\text{How}$ $\text{far}$ $\text{is}$ $\text{a}$ $\text{walk}$ $\text{around}$ $\text{it?}$

Answer Key

Level 3

1. $7\text{ and }8\text{ ft}$
2. $3\text{ and }4\text{ cm}$
3. $\approx 4\text{ m}$
4. $1\text{ cm}$
5. $\text{more } (\sqrt{90}>9)$
6. $64\text{ m}^3$
7. $\sqrt{2},\text{ irrational}$
8. $400\text{ m}$