Compare Methods for Simplifying Radicals

Jump to section:

Key Ideas Easy Medium Hard Super Challenge

What to Remember

Two useful paths

You can factor a perfect square (or cube) out first, or write a prime factorization and group pairs (or triples).

When factoring out is faster

If you see 36, 49, 8, or 27 right away, pull that factor and you are done.

When primes help

If the number is large or you are not sure, write prime factors. Pairs come out of a square root. Triples come out of a cube root.

Algebraic radicals

√x²y and ∛8a³ use the same laws. Pull perfect squares or cubes of the variables too.

Level 1

Easy

Show two methods on a small integer.

Example

Simplify √18 two ways: pull out 9, and use prime factors.

Way 1: 18 = 9 × 2, so √18 = 3√2.

Way 2: 18 = 2 × 3 × 3. One pair of 3s comes out: 3√2.

Same answer. Seeing 9 is faster here.

Answer: 3√2

Simplify √12 by pulling out 4, then by primes.

Show Answer

12 = 4 × 3 → 2√3

12 = 2 × 2 × 3 → 2√3

Answer: 2√3

Simplify ∛24 two ways.

Show Answer

24 = 8 × 3 → 2∛3

24 = 2³ × 3 → 2∛3

Answer: 2∛3

Which method is faster for √49?

Show Answer

49 is 7². Pulling out the perfect square is immediate: 7.

Answer: 7

Simplify √x² for x ≥ 0.

Show Answer

√x² = x

Answer: x

Simplify ∛8y³

Show Answer

8 = 2³, so 2y

Answer: 2y

Level 2

Medium

Choose a method and say why. Include a variable.

Example

Simplify √72. Name the method you used and one other method that also works.

72 = 36 × 2, so 6√2. Seeing 36 is fast.

Primes: 72 = 2³ × 3². One pair of 2s and one pair of 3s come out: 2 × 3 = 6, leftover 2.

Answer: 6√2

Simplify √20x² for x ≥ 0 two ways.

Show Answer

20x² = 4x² × 5 → 2x√5

Primes: 2² × 5 × x² → 2x√5

Answer: 2x√5

Simplify ∛54 with primes. Why is “pull out 27” also reasonable?

Show Answer

54 = 2 × 3³ → 3∛2

54 = 27 × 2 is the same 27 = 3³.

Answer: 3∛2

Simplify √48a³ for a ≥ 0.

Show Answer

48a³ = 16a² × 3a → 4a√(3a)

Answer: 4a√(3a)

A student always lists primes, even for √36. Is that wrong? Is it efficient?

Show Answer

Not wrong: 36 = 2² × 3² → 2 × 3 = 6.

Recognizing 36 = 6² is faster.

Answer: correct but slower

Simplify ∛27x³y

Show Answer

3x∛y

Answer: 3x∛y

Level 3

Hard

Compare methods on a messier expression.

Example

Simplify √180x⁴y for x ≥ 0, y ≥ 0. Show a perfect-square pull and a prime grouping.

180x⁴y = 36x⁴ × 5y → 6x²√(5y)

180 = 2² × 3² × 5, and x⁴ is (x²)². Same result.

Answer: 6x²√(5y)

Simplify ∛64a³b⁶c.

Show Answer

4ab²∛c

Answer: 4ab²∛c

Which method is better for √200: pull out 100, or primes? Do both.

Show Answer

200 = 100 × 2 → 10√2

200 = 2³ × 5² → 2 × 5 = 10 leftover 2. Pulling 100 is quicker if you see it.

Answer: 10√2

Simplify √12x² + √3x² for x ≥ 0. Why simplify each radical before adding?

Show Answer

2x√3 + x√3 = 3x√3

Without simplifying, the leftovers look different and you might think they cannot combine.

Answer: 3x√3

A student writes ∛16x³ = 2x∛8. Correct with a method of your choice.

Show Answer

16x³ = 8x³ × 2 → 2x∛2

Answer: 2x∛2

Explain when you would not use primes for √5².

Show Answer

√25 = 5 is immediate. Prime factors of 25 are 5 × 5, which is the same work with extra writing.

Answer: just 5

Super Challenge

Write a short comparison: for √8, √72, and √x²y³, which method would you use first, and why?

Show Answer

√8: 4 × 2 is obvious.

√72: 36 × 2 if you see 36; otherwise primes.

√x²y³: pull x² and y², leftover y. Variables make the perfect-power pull natural.

Answer: see comparison

Show that grouping primes in pairs (square root) vs triples (cube root) is the same law as √(a²b) = a√b and ∛(a³b) = a∛b.

Show Answer

A pair of equal primes is a square. A triple is a cube.

That square or cube is the a² or a³ you pull out.

Answer: pairs ↔ squares; triples ↔ cubes

Simplify √50x² − √8x² two ways: simplify each, or factor x² out of the whole expression first (x ≥ 0).

Show Answer

Each: 5x√2 − 2x√2 = 3x√2

Factor: x(√50 − √8) = x(5√2 − 2√2) = 3x√2

Answer: 3x√2

Need help choosing a radical method?

Book a Session with Amy