Number

Surds

Find exact lengths using square factors.

Predict, test and explain

Step 1 of 3

Predict

What is the exact side length of a square with area 72 cm²?

12=23\sqrt{12}=2\sqrt{3}
12 cm212\,\text{cm}^2
23 cm2\sqrt{3}\,\text{cm}

Scroll the diagram sideways to see every label.

side≈3.464 cm\text{side}\approx 3.464\,\text{cm}

The area is exact. Taking out a square factor gives an exact side length; the decimal is a rounded comparison.

Check the working
QuantityValue
Square factor4
Remaining radicand3
Exact side232\sqrt{3}

InvestigateWhich square factor can you take outside the root?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Simplify the surd, keeping an exact answer.

72\sqrt{72}

Hint

Split 72 into a square factor multiplied by another factor.

Reveal answer

Six root two. The square factor 36 comes out of the root as 6.

72=36×2=62\sqrt{72}=\sqrt{36\times2}=6\sqrt2

Build an explanation

Hint 1

Take square factors out of a square root: √18 = √(9 × 2) = 3√2.

Hint 2

Simplify each surd before collecting terms. Only terms with the same square-root part can be collected.

Hint 3

To rationalise a single-surd denominator, multiply numerator and denominator by that surd. Keep the answer exact.

Worked example

This example uses fixed values, separate from the diagram controls.

Simplify 2√18 − √8 + 1/√2 exactly.

  1. Simplify the square roots: 2√18 = 6√2 and √8 = 2√2.
  2. Rationalise: 1/√2 = √2/(√2 × √2) = √2/2.
  3. Collect the coefficients: 6√2 − 2√2 + √2/2 = (6 − 2 + 1/2)√2.

Answer: 9√2/2

Connect this idea

Watch out: √a + √b is not generally √(a + b). For example, √9 + √16 = 7, while √25 = 5.