Use square root and cube root symbols to represent solutions to equations of the form x = p and x = 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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Square roots and cube roots undo squaring and cubing. If a square has area 49 square units, its side is √49 = 7 units; if a cube has volume 64 cubic units, its edge is ∛64 = 4 units. Eighth graders use these symbols to write solutions to simple equations: x² = 81 has the solutions x = 9 and x = -9 (written x = ±√81), while x³ = 125 has the single solution x = 5.
Students are expected to know the perfect squares up to about 15² = 225 and the perfect cubes up to about 10³ = 1000 well enough to evaluate roots mentally, including roots of fractions such as √(9/16) = 3/4. The standard also asks them to know that √2 is irrational, connecting this work to 8.NS.A.1. When the number under the root is not a perfect square or cube, the exact answer stays in root form and a decimal is only an approximation.
For x² = 36, students give only x = 6. Remind them that (-6)² = 36 too, so the equation has two solutions, x = 6 and x = -6.
Students may say √16 = 8 or ∛27 = 9. Squaring 8 gives 64 and cubing 9 gives 729, which shows the slip immediately.
x³ = -8 has just one real solution, x = -2, because cubing a positive number can never give a negative result.
√(4/9) is 2/3, not 2/9. Both the numerator and the denominator are square-rooted.
Solve each equation: (a) x² = 144 (b) x³ = 216.
Answer: (a) x = 12 or x = -12. (b) x = 6.
Use square tiles and linking cubes: a 5 by 5 arrangement makes the link between 25 and √25 visible, and a 3 by 3 by 3 block does the same for ∛27. A running class chart of perfect squares and cubes, filled in by students, pays off across 8.G.B.7 and the volume work in 8.G.C.9.
Expect test items that pair an area or volume with a missing side, and equations with the solution set to choose from. Teach students to check every answer by squaring or cubing it, and to ask whether a negative value should be included.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 4
4 × 4 × 4 = 64, so the cube root of 64 is 4.
Answer: A) x = 7 and x = -7
Both 7² and (-7)² equal 49, so the solutions are x = 7 and x = -7.
Answer: 5 (also accepted: 5 cm)
Edge = ∛125 = 5, because 5 × 5 × 5 = 125.
Answer: C) √7
√81 = 9, ∛8 = 2 and √(1/4) = 1/2 are rational. 7 is not a perfect square, so √7 is irrational.
Answer: 4/5 (also accepted: 0.8)
√16 = 4 and √25 = 5, so √(16/25) = 4/5.
Answer: -3 (also accepted: x = -3, x=-3)
(-3) × (-3) × (-3) = -27, so x = -3. There is only one real solution.
Solve quadratic equations in one variable.
A full lesson with slides, activities and an exit ticket on square roots and cube roots, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.EE.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn square roots and cube roots into a quiz students answer online that marks itself, with a class summary for you.
Build a test →A positive and a negative number give the same square, but cubing keeps the sign. So squaring has two possible inputs for each positive output and cubing has only one.
No. Simplifying radicals belongs to high school. In grade 8 students evaluate perfect roots and leave others as √50 or estimate them.