How Do You Simplify The Square Root Of 72

Introduction

Simplifying the square root of 72 is a classic exercise that appears in middle‑school algebra, standardized‑test prep, and even everyday problem‑solving. While the expression √72 may look intimidating at first glance, it can be broken down into a product of simpler radicals that reveal its exact value in a more manageable form. Understanding how to simplify √72 not only strengthens your grasp of prime factorisation and the properties of radicals but also builds confidence for tackling more complex roots, rationalising denominators, and solving equations that involve surds. This guide walks you through every step, explains the underlying mathematics, and provides useful tips and practice problems to ensure you master the technique.

Why Simplify Radicals?

Before diving into the mechanics, it helps to know why we simplify radicals at all:

Clarity: A simplified radical such as 6√2 is easier to read and compare than √72.
Exactness: In many algebraic contexts, keeping the radical in its simplest form preserves the exact value, avoiding the rounding errors that come with decimal approximations.
Operations: Adding, subtracting, multiplying, or dividing radicals becomes straightforward when each term is expressed with the same radicand (the number under the root).
Standardisation: Textbooks, exams, and mathematical software expect radicals to be presented in their simplest form, so mastering the process is essential for academic success.

Step‑by‑Step Procedure for Simplifying √72

1. Prime Factorisation of the Radicand

The first step is to break the number under the radical (the radicand) into its prime factors.

72 = 2 × 36
    = 2 × 2 × 18
    = 2 × 2 × 2 × 9
    = 2 × 2 × 2 × 3 × 3

Thus, the prime factorisation of 72 is

72 = 2³ × 3².

2. Group the Factors in Pairs

A square root “cancels” a pair of identical factors because √(a²) = a. So, arrange the prime factors into pairs:

From 2³ we can form one pair of 2’s (2²) and leave a single 2 unpaired.
From 3² we have one complete pair of 3’s (3²).

So the factorisation can be written as

72 = (2²) × (3²) × 2.

3. Apply the Product Property of Square Roots

The product property states that √(ab) = √a · √b for non‑negative a and b. Using the paired and unpaired factors:

√72 = √[(2²) × (3²) × 2]
     = √(2²) × √(3²) × √2

4. Extract the Perfect Squares

Since √(k²) = k, the paired factors come out of the radical:

√(2²) = 2
√(3²) = 3

Putting it together:

√72 = 2 × 3 × √2 = 6√2

The simplified form of √72 is 6√2.

5. Verify (Optional)

To confirm, square the simplified result:

(6√2)² = 6² × (√2)² = 36 × 2 = 72

The verification shows that the simplification is correct.

Alternative Methods

Using the Largest Square Factor

Another quick way is to identify the largest perfect square that divides 72.

The squares less than 72 are 1, 4, 9, 16, 25, 36, 49, 64.
Among them, 36 is the greatest that divides 72 evenly (72 ÷ 36 = 2).

Write 72 as 36 × 2, then:

√72 = √(36 × 2) = √36 × √2 = 6√2

This method saves a few steps when you can spot the largest square factor instantly.

Using a Calculator for Approximation

If a decimal approximation is needed (e.g., for a physics problem), compute:

Continue exploring with our guides on why did victor create the monster in frankenstein and why are neodymium magnets so strong.

√72 ≈ 8.485281374...

But always keep the exact form 6√2 for algebraic work.

Common Mistakes to Avoid

Mistake	Why It’s Wrong	Correct Approach
Leaving a factor outside the radical without pairing it (e.g., writing √72 = 2√18)	2 is not a perfect square; the radical is not fully simplified.	Continue factoring until all perfect squares are extracted.
Forgetting to pair all possible factors (e.g., stopping at 2√18 instead of 6√2)	Results in a partially simplified expression.	Keep grouping factors until no more pairs remain.
Treating √a · √b = √(a + b)	Misapplies the product rule; addition is not allowed under the root.	Remember the correct rule: √a · √b = √(ab). Practically speaking,
Using negative numbers inside the square root without considering complex numbers	For real‑number simplifications, radicands must be non‑negative.	Ensure the radicand is non‑negative, or explicitly work in the complex plane.

Applications of the Simplified Form

Solving Equations – In equations like x² = 72, the solution is x = ±√72 = ±6√2, which is more informative than a decimal.
Geometry – The diagonal of a square with side length 6 is 6√2, derived directly from the Pythagorean theorem (√(6² + 6²)).
Physics – When calculating the magnitude of a vector with components (6, 6), the length is √(6² + 6²) = 6√2.
Financial Modelling – Certain risk formulas involve √(variance). If variance equals 72, the standard deviation is 6√2, a clean exact value useful for further symbolic manipulation.

Frequently Asked Questions

Q1: Can I simplify √72 to 8.5?

A: 8.5 is a rounded decimal approximation (8.485…). While acceptable for estimation, the exact simplified radical is 6√2. Use the exact form when precision matters.

Q2: What if the radicand contains a cube factor, like √108?

A: Factor 108 = 2² × 3³. Pair the squares (2²) and extract a 3 from the remaining 3³, leaving 3² × 3. The simplified form becomes 6√3.

Q3: Is there a shortcut for numbers that are multiples of 4?

A: Yes. Since 4 = 2² is a perfect square, you can factor out 4 first: √(4 × k) = 2√k. For 72, √(4 × 18) = 2√18, then continue simplifying √18 = 3√2, giving 6√2.

Q4: How does rationalising the denominator relate to simplifying √72?

A: If you encounter an expression like 1/√72, first simplify √72 to 6√2, then rationalise:

1/(6√2) = √2 / (6·2) = √2 / 12

Q5: Does the method change for higher roots, such as ∛72?

A: The principle is similar but uses cube factors. For ∛72, factor 72 = 2³ × 3². Extract the cube (2³) as 2, leaving ∛(3²) = ∛9. The simplified form is 2∛9.

Practice Problems

Simplify √50.
Write √98 in simplest radical form.
If x = 6√2, compute x².
Rationalise the denominator of 5 / √72.
Express √200 as a product of an integer and a simplified radical.

Answers:

5√2
7√2
72
5 / (6√2) → (5√2) / 12
10√2

Conclusion

Simplifying the square root of 72 is a straightforward process once you internalise the steps: factor the radicand, group factors into pairs, apply the product property of radicals, and extract perfect squares. Practically speaking, the result, 6√2, is not only compact but also reveals the exact relationship between 72 and the fundamental irrational number √2. Mastery of this technique equips you to handle a wide range of mathematical tasks—from algebraic equations and geometric calculations to physics problems and beyond. Keep practising with different numbers, recognise patterns such as the largest square factor, and you’ll find that simplifying radicals becomes an intuitive part of your mathematical toolkit.