Simplify and convert between mixed and entire radical forms. Math 10-C Alberta mathematics.
Lesson 2.3 of Number Systems & Operations in Math 10C — Alberta curriculum lessons.
Entire Radical. An entire radical is a radical with no number out front — everything sits under the radical sign.
Examples: √(80), √(54), √(32)
Mixed Radical. A mixed radical is a radical with a coefficient (a number multiplied out front) and a smaller radicand left under the radical sign.
Examples: 4√(5), 3√(2), 2√(2)
They Are the Same Value. Entire and mixed radicals are just two different ways of writing the same number.
Think of it like simplifying fractions — (3)/(12) and (1)/(4) are the same number, just written differently.
Similarly, √(80) and 4√(5) are the same number (both equal approximately 8.944).
Mixed radicals are the simplified form. They are easier to work with, and answers are usually expected in mixed radical form.
The Key Rule. The rule that makes converting between entire and mixed radicals possible is:
Multiplication Property of Radicals
√(a · b) = √(a) · √(b)
where n is a natural number, and a and b are real numbers.
This Works in Both Directions. Splitting: You can split a single radical into a product of two radicals.
Example: √(12) = √(4) · √(3)
Combining: You can combine two radicals into one.
Example: √(4) · √(3) = √(12)
This property is the foundation for converting between entire and mixed radicals.
Why You Need This List. To simplify radicals efficiently, you need to recognize perfect powers quickly.
A perfect square is a number like 4, 9, 16, 25 — numbers that are the square of a whole number.
A perfect cube is a number like 8, 27, 64, 125 — numbers that are the cube of a whole number.
Keep this reference handy as you work through simplification examples.
Example 1 — Simplify √(12). Step 1: What perfect square goes into 12? Answer: 4 (since 12 = 4 × 3).
Step 2: Split the radicand:
√(12) = √(4 × 3)
Step 3: Apply the multiplication property and simplify:
√(4 × 3) = √(4) · √(3) = 2√(3)
Check: (2)^2 × 3 = 4 × 3 = 12 ✓
Example 2 — Simplify √(24). Step 1: Largest perfect square in 24? Answer: 4 (since 24 = 4 × 6).
Step 2: Split:
√(24) = √(4 × 6)
Step 3: Simplify:
√(4) · √(6) = 2√(6)
Check: (2)^2 × 6 = 4 × 6 = 24 ✓
Example 3 — Simplify √(72). Step 1: Perfect squares that divide 72: 4, 9, 36. The largest is 36 (since 72 = 36 × 2).
Step 2: Split:
√(72) = √(36 × 2)
Step 3: Simplify:
√(36) · √(2) = 6√(2)
Check: (6)^2 × 2 = 36 × 2 = 72 ✓
What If You Miss the Largest Perfect Square?. You can simplify in stages — you still get the right answer, just with an extra step.
For √(72): if you started with √(72) = √(4 × 18) = 2√(18), notice that 18 still has a perfect square factor (9).
So: 2√(18) = 2 · √(9 × 2) = 2 · 3√(2) = 6√(2)
Same answer — but using the largest factor saves time.
Example 4 — Simplify √(128). Step 1: Largest perfect square in 128? Answer: 64 (since 128 = 64 × 2).
Step 2: Split:
√(128) = √(64 × 2)
Step 3: Simplify:
√(64) · √(2) = 8√(2)
Check: (8)^2 × 2 = 64 × 2 = 128 ✓
Example 5 — Simplify √(63). Step 1: Largest perfect square in 63? Answer: 9 (since 63 = 9 × 7).
Step 2: Split:
√(63) = √(9 × 7)
Step 3: Simplify:
√(9) · √(7) = 3√(7)
Check: (3)^2 × 7 = 9 × 7 = 63 ✓
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