8th Grade Square Roots and Cube Roots Lesson (Free PDF)
This square cube roots lesson for 8th Grade covers square roots and cube roots (Common Core 8.EE.A.2), with a free PDF to print.
A square root of p is a number that, when multiplied by itself, equals p; a cube root of p is a number whose cube equals p. The equation x squared equals p has two solutions, plus and minus the square root, while x cubed equals p has one real solution.
Grade 8 · Common Core State Standards: 8.EE.A.2
Key facts in this square cube roots lesson
- \(\sqrt{81} = 9\) 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.
What are square roots and cube roots?
A square root undoes squaring and a cube root undoes cubing. If a square has area 49 square inches, its side is the number that multiplies by itself to give 49.
Which squares and cubes should you know?
Know the squares up to \(15^{2} = 225\) and the cubes up to \(5^{3} = 125\). The table shows the squares and cubes from 1 to 10.
How do you solve x squared equals p?
Because both \(7 \cdot 7\) and \((-7) \cdot (-7)\) equal 49, the equation \(x^{2} = 49\) has two solutions, \(x = 7\) and \(x = -7\). We write \(x = \pm 7\). The equation \(x^{3} = 64\) has only one real solution, \(x = 4\), since a negative number cubed is negative.
- Isolate the power: for \(3x^{2} = 75\), divide by 3 to get \(x^{2} = 25\).
- Take the root: \(x = \pm\sqrt{25}\).
- State the answers: \(x = 5\) or \(x = -5\).
- Check: \(3 \cdot 25 = 75\).
What if the number is not a perfect square?
Then the root is irrational, so we leave it as \(\sqrt{p}\) or estimate it. For \(x^{2} = 20\), the solutions are \(x = \pm\sqrt{20}\). Since \(4^{2} = 16\) and \(5^{2} = 25\), the value is between 4 and 5, about 4.47. The number \(\sqrt{2}\) is irrational, so the side of a square with area 2 can never be written exactly as a fraction.
Fractions work too: \(\sqrt{\frac{9}{16}} = \frac{3}{4}\) and \(\sqrt[3]{\frac{8}{27}} = \frac{2}{3}\). Take the root of the numerator and of the denominator.
Square cube roots lesson: questions and answers
Why does x squared equal 49 have two answers?
Both 7 and -7 square to 49, because a negative times a negative is positive. So \(x = 7\) and \(x = -7\) both work.
Is the square root of a negative number real?
No. No real number squared is negative, so \(\sqrt{-9}\) is not a real number. A cube root of a negative number is fine: \(\sqrt[3]{-8} = -2\).
What does the radical symbol mean by itself?
The radical symbol means the positive square root only. So \(\sqrt{25} = 5\), while the equation \(x^{2} = 25\) has solutions 5 and -5.
How can I estimate a cube root?
Find the perfect cubes on each side. For \(\sqrt[3]{30}\), \(3^{3} = 27\) and \(4^{3} = 64\), so the root is a little more than 3.
Is the square root of 2 a fraction?
No. It is irrational: its decimal \(1.41421356…\) never ends or repeats, so no fraction equals it exactly.
How to use this square cube roots lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the square cube roots lesson as a PDF with the button below and keep it with your notes.
All Grade 8 chapters: Grade 8 math.
More 8th Grade math
Lessons
See all
Worksheets
See all
Tests
See all
Using Linear Models to Predict Test with Point Values- Slope and Similar Triangles Test with Point Values
- Rational and Irrational Numbers Test with Point Values
- Proportional Relationships and Slope Test with Point Values
- Understanding Functions Test with Point Values
- Properties of Integer Exponents Test with Point Values
