Using Surds in Exact Calculations
⭐ Higher Tier Content
Surds are exact values involving roots that cannot be written exactly as decimals. Using surds allows calculations to remain exact, avoiding rounding errors that occur when decimals are used too early.
In many mathematical problems, especially in algebra and geometry, answers are expected in exact form rather than as decimal approximations. This means leaving answers involving surds simplified, instead of converting them to decimals.
For example, the square root of \( 2 \) is an irrational number. Writing it as a decimal gives an approximation, but using the surd keeps the value exact.
$$
\sqrt{2}
$$
Exact calculations with addition and subtraction
Surds can be added or subtracted only when they are like surds.
For example:
$$
3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}
$$
This result is exact. Writing it as a decimal would introduce approximation.
Exact calculations with multiplication
When multiplying surds, multiply the numbers and then simplify.
For example:
$$
\sqrt{3} \times \sqrt{12}
$$
Simplify first:
$$
\sqrt{12} = 2\sqrt{3}
$$
So:
$$
\sqrt{3} \times 2\sqrt{3} = 2 \times 3
$$
$$
= 6
$$
This exact result is obtained without using any decimals.
Exact calculations in geometry
Surds often appear in geometric calculations, such as finding lengths using Pythagoras’ theorem. If the lengths of two sides of a right angled triangle are \( 3 \) cm and \( 4 \) cm, the hypotenuse is:
$$
\sqrt{3^2 + 4^2}
$$
$$
= \sqrt{25}
$$
$$
= 5
$$
If the sides are \( 1 \) cm and \( 1 \) cm, the hypotenuse is:
$$
\sqrt{1^2 + 1^2} = \sqrt{2}
$$
Leaving the answer as \( \sqrt{2} \) cm keeps it exact.
Why exact values matter
Using surds in exact calculations:
• avoids rounding errors
• keeps results accurate
• matches exam expectations
Decimals should only be used if the question specifically asks for an approximation. Otherwise, surd form is the preferred and most accurate way to present answers involving roots.