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Surds and square root rules

0:00A square has an area of twenty square centimetres, so its side length is √20 centimetres. A calculator gives a decimal approximation, but the square root itself tells us the exact length. We can also write that length as 2 × √5. By the end, you will understand this simplification and how to remove roots from denominators. We need three familiar ideas: square numbers, factors and equivalent fractions. A square number comes from multiplying a whole number by itself, such as nine or sixteen.

0:28Factors multiply together to make another number. Multiplying the numerator and denominator of a fraction by the same nonzero quantity leaves its value unchanged. A rational number can be written as a fraction of two integers, with a nonzero denominator. An irrational number cannot be written in that way. Its decimal continues forever without a repeating pattern. A surd is an irrational root of a rational number, and our examples will involve square roots.

0:53√9 is three, so it is not a surd. By contrast, √5 is irrational, which makes it a surd. For positive whole numbers, square roots of perfect squares are integers, while square roots of other whole numbers are irrational. The square root sign gives only the positive root, so √9 is three, not negative three. Keeping a surd preserves an exact value rather than replacing it with an approximation. Simplifying changes how that value is written, not the value itself.

1:20Our main tool is that the square root of a product equals the product of the square roots. In this video, the numbers inside the roots are always positive. Consider √4 × √5. Squaring this positive product gives 4 × 5, which is twenty. It must therefore be the positive √20. Since √4 is two, the product becomes 2 × √5.

Simplifying and combining surds

1:46For our first worked example, simplify √20. Write twenty as 4 × 5 because four is a perfect square. Separate the root into √4 × √5. Replace √4 with two, giving 2 × √5. The factor we want is a perfect square, because its root is a whole number. Choosing the biggest square factor saves extra steps. Five has no square factor other than one. So 2 × √5 is fully simplified.

2:15For another example, simplify √72. The largest perfect square factor is thirty-six, so write seventy-two as 36 × 2. Separate the roots to obtain √36 × √2. This gives 6 × √2. Adding surds works like collecting like terms in algebra. Three times the square root of two plus five times the square root of two gives 8 × √2. Both terms contain the same surd part, so we combine their coefficients.

2:45The coefficient is the number multiplying the surd. Now simplify the square root of two hundred minus the square root of thirty-two. Factor two hundred as 100 × 2, giving 10 × √2. For thirty-two, use 16 × 2 to obtain 4 × √2. The original expression therefore becomes ten times the square root of two minus four times the square root of two. Subtracting the coefficients gives 6 × √2.

3:13Simplifying first revealed that these were like terms. By contrast, the square root of two plus the square root of three cannot be collected into one term. Their surd parts are different. Addition combines matching quantities, while multiplication uses the product rule for square roots, so these operations need different treatment. For multiplication, √18 × √8 = √144. That simplifies to twelve, so multiplying two surds can produce a rational number.

3:41For division, √50 ÷ √2 becomes √25. The result is five, and the divisor must not be zero.

Rationalising surd denominators

3:50Rationalising the denominator means rewriting a fraction so that its denominator contains no surds. This does not make the entire fraction rational, because its numerator may still contain a surd. We preserve the fraction by multiplying its numerator and denominator by the same nonzero expression. The multiplier is chosen to turn the denominator into a rational number. Our next worked example is 10/√5. Multiply the numerator and denominator by √5.

4:17The numerator becomes 10 × √5, while the denominator becomes five. Dividing the coefficient ten by five leaves 2 × √5. This works because √5 multiplied by itself is five. We have multiplied the original fraction by one, written as √5/√5. Its value therefore stays unchanged. Multiplying only the denominator would change the fraction, so always show the matching multiplication in the numerator. A denominator containing two terms needs a different multiplier.

4:49For the square root of three minus one, use the square root of three plus one. These expressions are called conjugates because the sign between their terms changes. Multiplying them makes the mixed terms cancel, leaving a difference of squares. Multiply each term in the first expression by each term in the second expression. The square roots multiply to give three, and the remaining products are √3, minus √3, and negative one.

5:14Those opposite surd terms cancel. The result is 3 − 1, which equals two. Now rationalise one over negative one plus the square root of three. Reorder the denominator as the square root of three minus one. Multiply the numerator and denominator by its conjugate, the square root of three plus one. The numerator becomes the square root of three plus one. The denominator expands to three plus the square root of three, minus √3, minus one.

5:40Cancelling and subtracting gives the answer one plus the square root of three, all over two. Keep the whole numerator together when writing that final fraction. Both one and √3 are divided by two. Multiplying by the original denominator instead of its conjugate would leave surd terms after expansion. The conjugate works because the two mixed terms have equal sizes and opposite signs.

Common errors and exam skills

6:03One common mistake is splitting a square root across addition. The square root of nine plus sixteen is √25, which equals five. Separately taking the roots gives 3 + 4, which equals seven. These different results show why the product rule does not extend to addition or subtraction. Other common errors include combining unlike surds and missing signs during conjugate multiplication. Simplify each surd before deciding whether terms can be collected.

6:27Write every product when expanding a conjugate pair, including the final negative term. Also remember that 2 × √5 means multiplication, not √10. Exam questions may ask you to simplify an expression or rationalise its denominator, sometimes specifying an exact answer. These methods must work without a calculator on Paper two, and remain useful on calculator Paper four. An exact surd answer should not be replaced by a rounded decimal.

6:54If a decimal is requested, follow the stated accuracy. Otherwise use three significant figures, and avoid rounding intermediate values. The central idea is that surds let us keep irrational values exact. Simplification removes perfect square factors, and addition or subtraction combines matching surd parts. Rationalising preserves a fraction while making its denominator rational. For a single square root, multiply by that root; for a two-term denominator, use the conjugate so the mixed terms cancel.