Square Roots, Cube Roots, and Irrational Numbers

You will be able to:

• Use and to solve equations of the form and
• Evaluate roots of perfect squares, perfect cubes, and fractions fluently
• Classify square roots as rational or irrational

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Use to solve , distinguishing the principal root from
2. Use to represent the single real solution to
3. Evaluate square roots of perfect squares and cube roots of perfect cubes
4. Evaluate roots of perfect-square and perfect-cube fractions
5. Identify as irrational; classify roots of non-perfect-squares

What Is the Side Length of This Square?

Square with area 25 square units — what is the side length?

• side² = 25 → 5, because

Area = 36 → 6 · Area = 49 → 7 · Area = 1/4 → 1/2

Side lengths are positive — we want the positive answer.

Defining the Square Root: Principal Root

is the non-negative number whose square equals :

• · ·

The symbol always gives one value — the principal (positive) root.

Principal Root vs. Equation Solutions

• (the symbol → one value, always positive)
• Solutions to : (the equation → two values)

Solving x² = p Gives Two Solutions

(with ) has two solutions:

Both and square to .

• Symbol: (one value)
• Equation solutions:

Never write .

Perfect Squares and Fraction Roots Reference

Perfect squares 1² through 12²:

| 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |

Fraction rule:

· ·

Worked Example: Solving x² = 81/25

Solve

Step 1:

Step 2:

Step 3: Both solutions:

Step 4: Verify — ✓ and ✓

Guided Practice: Solve x² = 25/36

Solve

Step 1: ← complete this

Step 2: Write both solutions:

Step 3: Verify both answers by squaring.

(Complete steps 1–3 on your own.)

Check-In: Evaluate the Square Root, Then Solve

Evaluate

Then solve and state both solutions.

Think: what positive number, when squared, gives 9/16?

Practice: Three Square Root Problems

Solve each. Write all solutions and verify.

1. Evaluate
2. Solve
3. Solve

Practice Answers: Square Root Solutions

1. because

If you wrote : the symbol gives only the principal root. Use only for equation solutions.

From Square Roots to Cube Roots

Square roots undo squaring. Cube roots undo cubing.

Cube with volume 27 cubic units — what is the side length?

• side³ = 27 → 3, because

Volume = 8 → 2 · Volume = 125 → 5 · Volume = 1/8 → 1/2

Defining the Cube Root: One Real Solution

The cube root of , written , is the unique number whose cube equals :

Solving gives exactly one real solution:

Why Cube Root Equations Have One Solution

Cubing preserves sign — a negative number cubed is always negative.

Perfect Cubes and Fraction Cube Roots

Perfect cubes 1³ through 5³:

| | | | | |

Note: but — same number, different operations, different roots.

Fraction rule: · Example:

Worked Example: Solving x³ = 1/27

Solve

Step 1:

Step 2:

Step 3: Single solution: (no ±)

Step 4: Verify — ✓

Check-In: Compare Square and Cube Solutions

Solve both. State the number of solutions for each and explain why they differ.

Hint: 64 is both a perfect square () and a perfect cube ().

Practice: Three Cube Root Problems

Evaluate or solve. Write all solutions.

1. Evaluate
2. Evaluate
3. Solve

Practice Answers: Cube Root Solutions

1. because
2. · Verify: ✓
3. (one solution only)

If you wrote : does ? No — . One solution.

Square Roots of Non-Perfect-Squares Are Irrational

and — nice whole numbers. What about ?

• → is between 1 and 2
• Calculator:
• Never terminates. Never repeats. → irrational

No fraction satisfies — mathematicians have proven this.

Rational or Irrational? The Classification Table

The rule: is rational if and only if is a perfect square.

The Classification Rule with Examples

√2 Is Exact — Not an Approximation

Square with area 2 m²: , so m (exact).

• → is only an approximation
• The radical symbol names an exact irrational value

IS the precise number whose square is 2 — not "about 1.414."

Check-In: Classify Each Root as Rational or Irrational

Rational or irrational? If rational, give the exact value.

Practice Answers: Rational and Irrational Root Classification

1. → irrational
2. → rational
3. → irrational
4. → rational
5. → rational ()

Summary: Square Roots, Cube Roots, Irrational Roots

• : · : (one only)
• rational is a perfect square

Watch out:

• , not . Use only for equation solutions.
• Fractions work:
• but is irrational.
• : one solution. No ±.

Next Lesson: Approximating and Locating Irrational Roots

is irrational — but where does it sit on the number line?

Coming up in 8.NS.A.2:

• Locate , , between consecutive integers
• Narrow the interval: between 1.4 and 1.5, then 1.41 and 1.42

The exact values named today become the objects we locate tomorrow.