8.EE.A.2Common CoreMathExpressions and EquationsGrade 8
8.EE.A.2: Square Roots, Cube Roots and Irrational √2
In plain English: 8.EE.A.2 is the Common Core grade 8 math standard that asks students to use square root and cube root symbols to write the solutions of equations such as x² = p and x³ = p, to evaluate roots of small perfect squares and perfect cubes, and to know that √2 is irrational. It is taught in Grade 8 Math, in Expressions and Equations.
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Work with radicals and integer exponents. Also written as 8.EE.2 · Official standard
Students learn to undo squaring and cubing. A square root of a number p is a number that, multiplied by itself, gives p. A cube root of p is a number used as a factor three times to give p. Students use the root symbols √ and ∛ to write the solutions of equations such as x² = p and x³ = p, where p is a positive rational number (a number that can be written as a fraction of two integers, such as 5, 0.36 or 16/49). An equation x² = p has two solutions, √p and -√p, while x³ = p has one.
Students also find the roots of small perfect squares (squares of whole numbers, such as 144) and perfect cubes (cubes of whole numbers, such as 125), with tiles and linking cubes. The lesson ends with √2: students build a square with area 2, see that its side is √2, and find that no ending decimal squares to exactly 2. √2 is irrational: it cannot be written as a fraction of integers, and its decimal never ends or repeats.
Learning Objectives
By the end of this lesson, students will be able to:
Write the solutions of x² = p as x = √p or x = -√p, and the solution of x³ = p as x = ∛p, for a positive rational number p
Evaluate square roots of small perfect squares and cube roots of small perfect cubes, including fractions and decimals built from them
Explain why x² = p has two solutions but x³ = p has only one
Explain that √2 is irrational and that decimals such as 1.414 are only close to it
Prior Knowledge Required
Students should already be comfortable with:
Evaluating whole-number powers such as 7² and 4³ 6.EE.A.1
Finding the area of a rectangle by multiplying side lengths 3.MD.C.7
Finding the volume of a rectangular prism, such as a cube, by multiplying edge lengths 5.MD.C.5
Knowing that the decimal form of a rational number ends or repeats 7.NS.A.2
Write the prompt on the board. Remind students that 5² means 5 × 5 (5 squared) and 5³ means 5 × 5 × 5 (5 cubed). Give pairs three minutes.
Warm-Up Prompt
"Find a number that you can multiply by itself to get 64. Can you find a second one? Then find a number you can use as a factor three times to get 64."
Collect answers. Most pairs find 8, because 8 × 8 = 64. Ask whether a negative number could work: (-8) × (-8) = 64 as well, since a negative times a negative is positive. For the second question, 4 × 4 × 4 = 64. Tell students that 8 and -8 are the square roots of 64 and that 4 is its cube root. Today they learn symbols for these numbers and use them to solve equations.
Direct Instruction20 minutes
Write each idea on an anchor chart, with the warm-up numbers as the first example:
Square root: a number that, multiplied by itself, gives p. Every positive number has two square roots, one positive and one negative. The square roots of 64 are 8 and -8.
The symbol √ (the radical sign) always means the positive square root. √64 = 8. To name the negative one, write -√64 = -8.
Cube root: a number used as a factor three times to give p, written ∛p. ∛64 = 4. A positive number has one cube root, because a negative number cubed is negative: (-4)³ = -64.
Solving x² = p: x = √p or x = -√p, written x = ±√p and read "plus or minus the square root of p." Solving x³ = p: x = ∛p.
Perfect squares and perfect cubes: 1, 4, 9, 16, 25, ... are the squares of whole numbers, and 1, 8, 27, 64, ... are the cubes. Their roots are whole numbers. The root of a fraction built from them is a fraction: take the root of the top and of the bottom.
When p is not a perfect square or cube, leave the answer with the symbol, such as x = ±√7. This is an exact answer. Decimals such as 2.65 are only close to √7.
Solving x² = p with a perfect square
Solve x² = 81.
Equation: x = √81 or x = -√81, so x = 9 or x = -9. Check: 9² = 81 and (-9)² = 81
Solving x³ = p with a perfect cube
Solve x³ = 125.
Equation: x = ∛125 = 5, because 5 × 5 × 5 = 125. There is no second solution: (-5)³ = -125
A fraction under the square root
Solve x² = 16/49.
Equation: x = ±√(16/49) = ±4/7, because (4/7)² = 16/49 and (-4/7)² = 16/49
A fraction under the cube root
Solve x³ = 27/64.
Equation: x = ∛(27/64) = 3/4, because 3³ = 27 and 4³ = 64
An answer that stays with the symbol
A square has an area of 2 square units. Find its side length s.
Equation: s² = 2, so s = √2 units (the negative solution -√2 cannot be a length). √2 is irrational, so no fraction or ending decimal is exactly equal to it; 1.414 is only close
Use Diagram 2 to connect the words to pictures: a square with 25 unit squares has side √25 = 5, and a cube with 27 unit cubes has edge ∛27 = 3. Then use Diagram 1 for the last example. The shaded square is made of four half-squares, so its area is 2, and its side is √2. Explain that √2 is irrational: it is not the ratio (fraction) of two integers. Tell students they will test this themselves in Activity 3, and that mathematicians have proved that no fraction at all has a square of exactly 2.
Guided Practice15 minutes
Pairs copy and complete the table. After each row, one pair explains how many solutions the equation has and why.
Write the solutions with a root symbol, then simplify if you can.
Equation
Solutions with a root symbol
Simplified
x² = 144
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x² = 0.36
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x³ = 8/27
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x³ = 1
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x² = 7
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Answers: x = ±√144 = ±12; x = ±√0.36 = ±0.6, because 0.6 × 0.6 = 0.36; x = ∛(8/27) = 2/3; x = ∛1 = 1; x = ±√7, which stays with the symbol because 7 is not a perfect square. Then fix an error together: Dev writes √16 = ±4. The symbol √16 names only the positive root, 4. It is the equation x² = 16 that has two solutions, 4 and -4.
Independent Practice15 minutes
Students work alone on five problems, then compare with a partner. (1) Solve x² = 121. (x = ±11.) (2) Solve x³ = 216. (x = 6.) (3) Solve x² = 1/4. (x = ±1/2.) (4) Solve x³ = 0.125. (x = 0.5, because 0.5 × 0.5 × 0.5 = 0.125.) (5) Solve x² = 10 and explain why you leave the answer with the symbol. (x = ±√10; 10 is not a perfect square, since 3² = 9 and 4² = 16.)
Closure5-10 minutes
Exit ticket: (1) Solve x² = 100. (x = 10 or x = -10.) (2) Evaluate ∛8. (2.) (3) A classmate says √2 = 1.414. Is that exactly right? Explain. (No: 1.414 × 1.414 = 1.999396, which is not 2. √2 is irrational, so 1.414 is only an approximation.)
Differentiation Strategies
For Struggling Students
Give a reference card with the squares of 1 to 12 and the cubes of 1 to 5, and have students circle p on it before they solve
Have students build the square or cube with tiles or linking cubes when p is small, then write the root
Ask students to check every solution by squaring or cubing it, including the negative one
For Advanced Students
Ask why x² = 0 has only one solution while x² = 9 has two
Ask students to find all whole numbers from 1 to 100 that are both perfect squares and perfect cubes (1 and 64) and predict the next one (they are sixth powers, so the next is 3⁶)
Extension (beyond this standard): show the classic proof that no fraction a/b squares to 2. If a/b is in lowest terms and a² = 2b², then a is even, so b is even too, which contradicts lowest terms
Assessment Guidance
What to Look For
Check that students write both solutions of x² = p and only one of x³ = p, and that they check solutions by squaring or cubing. Watch for four errors: dividing p by 2 or 3 instead of taking a root, writing √16 = ±4, taking the root of only the top of a fraction, and calling 1.414 the exact value of √2. A strong answer says why a root is exact (the square of the answer is p) or why the answer must stay with the symbol.
02
Classroom Activities
3 Activities
1
Tile Squares and Cube Towers
15 minGroups of 3
Groups use square tiles to build squares and linking cubes to build cubes. They record which numbers of tiles make a square and which numbers of cubes make a cube, and write each side or edge with a root symbol.
Tasks
Build a solid square with 4, 9, 16, 25 and 36 tiles. Record each side, for example √16 = 4
Try to build a solid square with 12 tiles and with 20 tiles, and explain what goes wrong
Build a solid cube with 8 linking cubes and with 27 linking cubes. Record each edge, for example ∛8 = 2
Try to build a solid cube with 12 linking cubes
Procedure
One student builds, one records, one checks by counting rows (squares) or layers (cubes); rotate roles for each number
For each number that fails, write the two perfect squares or cubes it lies between, for example 9 < 12 < 16
Discussion Questions
Six numbers from 1 to 36 are perfect squares. Which are they?
From 1 to 36, only one number is both a perfect square and a perfect cube. Which one?
Why can't 12 tiles make a solid square even though 12 is even?
Modification for Distance Learning
Use virtual tiles on a shared grid, or have students shade squares on grid paper and draw cubes layer by layer.
2
Build a Square with Area 2
20 minPairs
Each student gets two paper squares, 5 cm on a side. Call the side of one paper square 1 unit, so each paper square has an area of 1 square unit. Students cut and rearrange the two squares into one square with an area of 2 square units, as in Diagram 1.
Procedure
Cut each paper square along one diagonal to make 4 right triangles
Arrange the 4 triangles into one square with the right angles meeting at the center, then tape them
Explain why the new square has an area of 2 square units and write its side length with a root symbol (√2 units)
Measure the side with a centimeter ruler. The side is about 7.1 cm. Divide by 5 cm to write it in units (about 1.42 units)
Sample Measurements (invented)
An invented class of pairs measured 7.0 cm, 7.1 cm, 7.1 cm, 7.2 cm and 7.1 cm.
Discussion Questions
Every pair measured about 7.1 cm, but not exactly the same. Could a better ruler give the exact length as a decimal?
Why is the side of the new square the same as the diagonal of one paper square?
Which equation does the side length s solve?
3
The √2 Decimal Hunt
15 minPairs
Pairs try to find a decimal whose square is exactly 2, using a calculator with the square root key covered. They only multiply. The hunt shows that the squares get closer and closer to 2 but never land on it.
The Hunt (with answers)
1² = 1 and 2² = 4, so √2 is between 1 and 2
1.41² = 1.9881 and 1.42² = 2.0164, so √2 is between 1.41 and 1.42
1.414² = 1.999396 and 1.415² = 2.002225, so √2 is between 1.414 and 1.415
Pairs continue with 4 decimal places as far as they can
Why No Ending Decimal Works
Look at the last digit. The squares of the digits 1 to 9 end in 1, 4, 9, 6, 5, 6, 9, 4 and 1, never in 0. So if a decimal ends in a nonzero digit, its square has twice as many decimal places and its last digit is not 0. That square can never be exactly 2 = 2.000... . Together with the fact that no fraction squares to 2, this is why √2 is irrational.
Discussion Questions
1.414 has 3 decimal places. How many decimal places does its square have?
Why is 1.414 called an approximation of √2?
Is √4 irrational too? Explain.
Challenge Variation
Repeat the hunt for ∛2, cubing instead of squaring. Is ∛2 between 1.2 and 1.3?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Square with Area 2 Has Side √2
Each small square has an area of 1 square unit. The shaded square is made of four half-squares, so its area is 2 square units and its side length is √2 units, the diagonal of one small square.
Diagram 2: Square Roots Give Sides, Cube Roots Give Edges
A square made of 25 unit squares has side √25 = 5. A cube made of 27 unit cubes, 3 layers of 9, has edge ∛27 = 3.
04
Homework Assignment
~30 min
8.EE.A.2 Homework: Square Roots, Cube Roots and √2
Directions: Write every solution with a root symbol first, then simplify it if you can. Check each answer by squaring or cubing it. Do not use the square root key on a calculator.
(a) A square floor tile has an area of 1,600 cm². How long is each side? (b) A cube-shaped moving box has a volume of 125,000 cm³. How long is each edge? (c) A square garden has an area of 20 m². Write its exact side length with a root symbol, and explain why it is not a whole number of meters.
(a) Explain why x² = 81/100 has two solutions, and give both. (b) Explain why x³ = 1,331 has only one solution, and give it.
Sam says √2 = 17/12 because (17/12)² is very close to 2. (a) Find (17/12)² as a fraction. (b) Is Sam right? Explain. (c) What kind of number is √2, and what does that tell you about any fraction someone suggests for it?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Root Symbols
Every solution is first written with √ or ∛, including ±√p for x² = p
Symbols used for some solutions
Symbols missing or misused
Evaluating Roots
All roots of perfect squares, perfect cubes and their fractions and decimals are correct
One or two roots are wrong
Most roots are wrong
Number of Solutions
Gives two solutions for x² = p and one for x³ = p, with a reason
Correct count without a reason
Wrong count
√2 Is Irrational (Problem 6)
Shows that (17/12)² is not 2 and explains that no fraction equals √2
Shows the square but the explanation is incomplete
No correct work
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order without the square root key on a calculator. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What is the value of √196?
Answer: B
14 × 14 = 196, and the symbol √ names the positive square root, so √196 = 14. Choice A divides 196 by 2. Choice C gives both square roots, but √196 means only the positive one. Choice D moves the decimal point instead of taking a root.
Question 2 of 20 · Multiple Choice
What is the value of ∛729?
Answer: D
9 × 9 × 9 = 729, so ∛729 = 9. Choice A divides 729 by 3. Choice B is the square root of 729, since 27 × 27 = 729. Choice C uses 9 × 81 = 729 and picks the wrong factor; 81 × 81 × 81 is far more than 729.
Question 3 of 20 · Multiple Choice
Which gives all the solutions of x² = 0.49?
Answer: A
0.7 × 0.7 = 0.49 and (-0.7) × (-0.7) = 0.49, so there are two solutions. Choice B misses the negative solution. Choice C has the decimal point in the wrong place: 0.07² = 0.0049. Choice D divides 0.49 by 2.
Question 4 of 20 · Multiple Choice
Which gives all the solutions of x³ = 1/64?
Answer: C
(1/4)³ = 1/64, so x = ∛(1/64) = 1/4. Choice A adds a negative solution, but (-1/4)³ = -1/64. Choice B is the square root of 1/64. Choice D divides by 3 instead of taking a cube root.
Question 5 of 20 · Multiple Choice
Which equation has the solutions x = √13 and x = -√13?
Answer: A
The solutions of x² = p are √p and -√p, so p = 13. Choice B has only one solution, ∛13. Choice C squares 13 again: its solutions are 13 and -13. Choice D doubles 13, as if squaring meant multiplying by 2.
Question 6 of 20 · Multiple Choice
Which statement about √2 is true?
Answer: C
An irrational number cannot be written as a fraction of two integers, and no such fraction squares to 2. Choice A is false: 1.414 is only close to √2. Choice B uses a fraction whose top is not an integer. Choice D is false: √4 = 2 is rational.
Question 7 of 20 · Multiple Choice
Ana writes √2 = 7/5, because 7/5 = 1.4 and √2 starts with 1.4. Which statement is correct?
Answer: B
To test a square root, square it: 49/25 = 1.96, not 2. Choice A trusts the first digits, but many numbers start with 1.4. Choice C is false: 7/5 is a fraction of integers, so it is rational. Choice D is false: no whole number squares to 2, since 1² = 1 and 2² = 4.
Question 8 of 20 · Multiple Choice
A square window pane has an area of 400 square inches. How long is each side?
Answer: D
The side s solves s² = 400, and s = √400 = 20 inches, since 20 × 20 = 400. Choice A divides by 2. Choice B divides by 4, mixing up area with perimeter. Choice C squares 400 instead of taking its square root.
Question 9 of 20 · Multiple Choice
A cube-shaped planter holds 27,000 cm³ of soil when full. How long is each edge?
Answer: A
The edge e solves e³ = 27,000. 30 × 30 × 30 = 27,000, so e = ∛27,000 = 30 cm. Choice B divides by 3. Choice C takes ∛27 = 3 but keeps all three zeros; 3,000 cubed is far more than 27,000. Choice D is 30², the area of one face, not the edge.
Question 10 of 20 · Multiple Choice
Which gives all the solutions of x² = 50?
Answer: C
The solutions of x² = p are √p and -√p. 50 is not a perfect square, so the answers stay with the symbol. Choice A misses the negative solution. Choice B divides 50 by 2. Choice D uses cube roots, which solve x³ = 50.
Question 11 of 20 · Multiple Choice
Which number is a perfect square?
Answer: D
16 × 16 = 256, so √256 = 16. The others fall between perfect squares: 11² = 121 < 128 < 144 = 12², 12² = 144 < 160 < 169 = 13², and 15² = 225 < 250 < 256. A student may pick 128 because 128 = 2 × 64 and 64 is a perfect square, but doubling a perfect square does not give one.
Question 12 of 20 · Multiple Choice
What is the value of ∛(8/125)?
Answer: B
∛8 = 2 and ∛125 = 5, so ∛(8/125) = 2/5. Check: (2/5)³ = 8/125. Choice A takes the cube root of the top only. Choice C is negative, but (-2/5)³ = -8/125. Choice D divides by 3: 125 × 3 = 375.
Question 13 of 20 · Multiple Choice
Which statement about the equation x³ = 343 is true?
Answer: A
7 × 7 × 7 = 343, and a negative number cubed is negative, so 7 is the only solution. Choice B treats the cube like a square. Choice C uses a factor pair of 343 instead of three equal factors. Choice D confuses perfect squares with perfect cubes: 343 is a perfect cube.
Question 14 of 20 · Multiple Choice
The equation x² = p has the solutions x = 1/3 and x = -1/3. What is p?
Answer: C
p is the square of either solution: (1/3)² = 1/9, and (-1/3)² = 1/9 too. Choice A doubles 1/3 instead of squaring it. Choice B halves 1/3. Choice D thinks the negative solution makes p negative, but a negative number squared is positive.
Question 15 of 20 · Short Answer
Solve x² = 25/64. Write the solutions with a root symbol, then simplify.
x = ±√(25/64). Since 5² = 25 and 8² = 64, x = 5/8 or x = -5/8.
Question 16 of 20 · Short Answer
Solve x³ = 0.001.
x = ∛0.001. Since 0.1 × 0.1 × 0.1 = 0.001, x = 0.1. It is the only solution, because (-0.1)³ = -0.001.
Question 17 of 20 · Short Answer
Square 1.4142 by hand or with the multiplication key. Is 1.4142 exactly √2? Explain.
1.4142 × 1.4142 = 1.99996164. No: the square is close to 2 but not equal to it. √2 is irrational, so every ending decimal, including 1.4142, is only an approximation.
Question 18 of 20 · Short Answer
A square rug has an area of 6 square meters. Write the exact length of one side with a root symbol. The equation s² = 6 has two solutions; why does only one of them fit?
The side is √6 meters (a little less than 2.5 m). The equation s² = 6 also has the solution -√6, but a length cannot be negative, so only √6 fits.
Question 19 of 20 · Short Answer
Solve both equations and explain the difference in the number of solutions: (a) x² = 11 (b) x³ = 11.
(a) x = √11 or x = -√11, because a positive or a negative number squared is positive. (b) x = ∛11 only, because a negative number cubed is negative, so it cannot equal 11. Neither answer simplifies, since 11 is not a perfect square or a perfect cube.
Question 20 of 20 · Short Answer
Is each number rational or irrational? Explain each answer: √2, √(25/4), 1.41.
√2 is irrational: no fraction of integers equals it. √(25/4) is rational: it equals 5/2. 1.41 is rational: it equals 141/100. It is close to √2 but not equal, since 1.41² = 1.9881.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.A.2 mean?
8.EE.A.2 means students use the symbols √ and ∛ to write the solutions of x² = p and x³ = p, evaluate the roots of small perfect squares and cubes, and know that √2 is irrational. For example, x² = 36 has the solutions 6 and -6, and x³ = 8 has the solution 2.
Why does x² = p have two solutions but x³ = p only one?
Because squaring hides the sign but cubing keeps it. A positive number and its opposite have the same square, so both are solutions of x² = p. A negative number cubed is negative, so only the positive cube root solves x³ = p when p is positive.
Does √9 equal 3 or ±3?
√9 = 3. The radical sign always names the positive square root. The equation x² = 9 has two solutions, 3 and -3, which is why we write x = ±√9 when we solve it.
What counts as a small perfect square or perfect cube in 8.EE.A.2?
The standard does not give a list. Classes usually learn the squares of 1 to 15 and the cubes of 1 to 10, such as 144 and 125. Fractions and decimals built from them, such as 9/100 or 0.008, have roots students can find the same way.
What does it mean that √2 is irrational?
It means √2 cannot be written as a fraction of two integers. Its decimal, 1.41421356..., never ends and never repeats. Decimals such as 1.415 and fractions such as 99/70 are only approximations.
Do students need to prove that √2 is irrational?
No. The standard says students should know it. Many classes show why no ending decimal works by squaring decimals, as in this lesson's Activity 3. The full proof that no fraction works is usually shown as an extension, in grade 8 or in high school.
Can p be a fraction or a decimal?
Yes. The standard says p is a positive rational number. For example, x² = 36/121 has the solutions 6/11 and -6/11, and x³ = 0.008 has the solution 0.2. If p is not built from perfect squares or cubes, leave the answer with the symbol.
What mistakes do students make with square and cube roots?
A few come up often: dividing p by 2 or 3 instead of taking a root, forgetting the negative solution of x² = p, adding a negative solution to x³ = p, and treating 1.414 as the exact value of √2. Checking by squaring or cubing the answer catches each of these.
How does 8.EE.A.2 connect to other standards?
It works alongside 8.NS.A.1, where students learn that numbers like √2 are irrational, and 8.EE.A.1 on exponents. Next, students use square roots to find side lengths with the Pythagorean Theorem (8.G.B.7). In high school, HSN.RN.B.3 explains why sums and products with irrational numbers behave as they do.
How can parents help with square and cube roots at home?
Use floor tiles or a chessboard: an 8 by 8 board has 64 squares, so its side is √64 = 8 squares. Ask your child for the side of a square garden with a given area, and for the edge of a cube-shaped box with a given volume. Have them check each answer by multiplying.
07
Related Standards
5 standards
These standards connect to 8.EE.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.EE.A.1Prerequisite
Write and evaluate numerical expressions with whole-number exponents
