Radical Expressions
📖 Learn
What is a Radical Expression?
A radical expression is any mathematical expression containing a radical symbol (root symbol). The most common is the square root, but we also work with cube roots, fourth roots, and beyond. The general form is: n-th root of a, written as the radical symbol with index n and radicand a.
Key Vocabulary
| Term | Definition | Example |
|---|---|---|
| Radical | The root symbol and its contents | The square root of 25 |
| Radicand | The number or expression under the radical | In the square root of 25, the radicand is 25 |
| Index | The small number indicating the root type | In the cube root of 8, the index is 3 |
| Perfect Square | A number whose square root is a whole number | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| Perfect Cube | A number whose cube root is a whole number | 1, 8, 27, 64, 125 |
Simplifying Radicals
A radical is in simplest form when:
- No perfect square factors remain under the radical (for square roots)
- No fractions are under the radical
- No radicals are in the denominator
The square root of (a times b) equals the square root of a times the square root of b, where a and b are greater than or equal to 0.
This allows us to split radicals: the square root of 50 = the square root of (25 times 2) = the square root of 25 times the square root of 2 = 5 times the square root of 2.
The square root of (a divided by b) equals the square root of a divided by the square root of b, where a is greater than or equal to 0 and b is greater than 0.
Adding and Subtracting Radicals
You can only add or subtract radicals that have the same index AND the same radicand (called like radicals).
Example: 3 times the square root of 5 plus 7 times the square root of 5 = 10 times the square root of 5
But: 3 times the square root of 5 plus 2 times the square root of 3 cannot be combined (different radicands)
Multiplying Radicals
To multiply radicals with the same index, multiply the radicands:
The square root of a times the square root of b = the square root of (a times b)
Rationalizing the Denominator
When a radical appears in a denominator, we rationalize by multiplying both numerator and denominator by an appropriate radical to eliminate the radical from the denominator.
Conjugates
Conjugates are expressions that differ only in the sign between two terms. For example, (a + the square root of b) and (a - the square root of b) are conjugates. When multiplied together, they eliminate the radical: (a + the square root of b)(a - the square root of b) = a squared minus b.
💡 Examples
Example 1: Simplifying a Square Root
Problem: Simplify the square root of 72
Solution:
Step 1: Find the largest perfect square factor of 72
72 = 36 times 2, and 36 is a perfect square
Step 2: Apply the product property
The square root of 72 = the square root of 36 times the square root of 2
Step 3: Simplify
Answer: 6 times the square root of 2
Example 2: Adding Radicals
Problem: Simplify: 2 times the square root of 18 + 3 times the square root of 8
Solution:
Step 1: Simplify each radical first
The square root of 18 = the square root of 9 times the square root of 2 = 3 times the square root of 2
The square root of 8 = the square root of 4 times the square root of 2 = 2 times the square root of 2
Step 2: Substitute back
2(3 times the square root of 2) + 3(2 times the square root of 2)
= 6 times the square root of 2 + 6 times the square root of 2
Step 3: Combine like radicals
Answer: 12 times the square root of 2
Example 3: Multiplying Radicals
Problem: Multiply: (the square root of 3)(the square root of 15)
Solution:
Step 1: Multiply the radicands
The square root of (3 times 15) = the square root of 45
Step 2: Simplify the result
The square root of 45 = the square root of 9 times the square root of 5 = 3 times the square root of 5
Answer: 3 times the square root of 5
Example 4: Rationalizing a Simple Denominator
Problem: Rationalize: 5 divided by the square root of 3
Solution:
Step 1: Multiply numerator and denominator by the square root of 3
= (5 times the square root of 3) divided by (the square root of 3 times the square root of 3)
Step 2: Simplify
= (5 times the square root of 3) divided by 3
Answer: (5 times the square root of 3) divided by 3
Example 5: Rationalizing Using Conjugates
Problem: Rationalize: 6 divided by (2 + the square root of 5)
Solution:
Step 1: Multiply by the conjugate (2 - the square root of 5) over itself
= 6(2 - the square root of 5) divided by (2 + the square root of 5)(2 - the square root of 5)
Step 2: Apply difference of squares in denominator
= (12 - 6 times the square root of 5) divided by (4 - 5)
= (12 - 6 times the square root of 5) divided by (-1)
Step 3: Simplify
Answer: -12 + 6 times the square root of 5 (or 6 times the square root of 5 minus 12)
✏️ Practice
Try these problems on your own. Choose the best answer for each question.
1. Simplify: the square root of 98
A) 7 times the square root of 2
B) 49 times the square root of 2
C) 2 times the square root of 49
D) 14 times the square root of 2
2. Simplify: the square root of 45 + the square root of 20
A) the square root of 65
B) 5 times the square root of 5
C) 7 times the square root of 5
D) 3 times the square root of 5 + 2 times the square root of 5
3. Multiply: (2 times the square root of 6)(3 times the square root of 2)
A) 6 times the square root of 12
B) 12 times the square root of 3
C) 5 times the square root of 8
D) 6 times the square root of 8
4. Rationalize: 8 divided by the square root of 2
A) 4 times the square root of 2
B) 8 times the square root of 2
C) 4 divided by the square root of 2
D) (8 times the square root of 2) divided by 2
5. Simplify: the square root of 12 times the square root of 3
A) the square root of 36
B) 6
C) 3 times the square root of 4
D) Both A and B
6. Which expression is in simplest form?
A) the square root of 50
B) 3 divided by the square root of 5
C) 2 times the square root of 7
D) the square root of (1/4)
7. Simplify: 5 times the square root of 32 minus 2 times the square root of 18
A) 3 times the square root of 14
B) 20 times the square root of 2 minus 6 times the square root of 2
C) 14 times the square root of 2
D) 26 times the square root of 2
8. Rationalize: 3 divided by (1 + the square root of 2)
A) 3 minus 3 times the square root of 2
B) -3 + 3 times the square root of 2
C) 3 plus 3 times the square root of 2
D) (3 + 3 times the square root of 2) divided by 3
9. Simplify the cube root of 54
A) 3 times the cube root of 2
B) 2 times the cube root of 3
C) 27 times the cube root of 2
D) 6 times the cube root of 9
10. If the square root of x = 5 times the square root of 2, what is x?
A) 10
B) 25
C) 50
D) 100
Click to reveal answers
- A) 7 times the square root of 2 — 98 = 49 times 2, so the square root of 98 = 7 times the square root of 2
- B) 5 times the square root of 5 — The square root of 45 = 3 times the square root of 5, the square root of 20 = 2 times the square root of 5; sum = 5 times the square root of 5
- B) 12 times the square root of 3 — 2 times 3 = 6; the square root of 6 times the square root of 2 = the square root of 12 = 2 times the square root of 3; 6 times 2 = 12
- A) 4 times the square root of 2 — Multiply by the square root of 2 over the square root of 2: (8 times the square root of 2) divided by 2 = 4 times the square root of 2
- D) Both A and B — The square root of 12 times the square root of 3 = the square root of 36 = 6
- C) 2 times the square root of 7 — No perfect square factors, no fraction under radical, no radical in denominator
- C) 14 times the square root of 2 — 5(4 times the square root of 2) - 2(3 times the square root of 2) = 20 times the square root of 2 - 6 times the square root of 2 = 14 times the square root of 2
- B) -3 + 3 times the square root of 2 — Multiply by (1 - the square root of 2)/(1 - the square root of 2): 3(1 - the square root of 2)/(1 - 2) = (3 - 3 times the square root of 2)/(-1)
- A) 3 times the cube root of 2 — 54 = 27 times 2 = 3 cubed times 2, so the cube root = 3 times the cube root of 2
- C) 50 — Square both sides: x = 25 times 2 = 50
✅ Check Your Understanding
Question 1: Why can we only combine like radicals when adding or subtracting?
Reveal Answer
Like radicals have the same index and radicand, which means they represent the same irrational value. Just as we can only combine like terms in algebra (3x + 2x = 5x, but 3x + 2y cannot be simplified), we can only combine radicals that represent the same type of quantity. For example, 3 times the square root of 5 + 2 times the square root of 5 = 5 times the square root of 5 because both terms are multiples of the same irrational number (the square root of 5).
Question 2: Why do we need to rationalize the denominator?
Reveal Answer
Historically, rationalizing made calculations easier before calculators. Today, we still rationalize because: (1) It is the conventional "simplified" form expected in mathematics; (2) It makes comparing expressions easier; (3) It helps in further algebraic manipulations; (4) Standardized tests like the SAT/ACT expect answers in rationalized form. Rationalizing transforms an expression into an equivalent form that's often more useful for further work.
Question 3: Explain how the conjugate method eliminates radicals from the denominator.
Reveal Answer
When we multiply a binomial containing a radical by its conjugate, we apply the difference of squares pattern: (a + b)(a - b) = a squared minus b squared. For example, (3 + the square root of 5)(3 - the square root of 5) = 9 - 5 = 4. The radical terms multiply to give a perfect square (the square root of 5 times the square root of 5 = 5), and the opposite signs cause the middle terms to cancel. This leaves only rational numbers in the denominator.
Question 4: What is the first step you should always take when simplifying a radical?
Reveal Answer
The first step is to find the largest perfect square (for square roots) or perfect cube (for cube roots) factor of the radicand. Finding the LARGEST perfect square factor is key because it ensures you fully simplify in one step. For example, with the square root of 72, you could factor as the square root of (4 times 18) = 2 times the square root of 18, but you'd need to simplify again. Using the square root of (36 times 2) = 6 times the square root of 2 completes the simplification immediately.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review