Exact Values (Edexcel GCSE Maths: Foundation): Revision Note

Exact values

What are exact values?

• An exact value is one where the answer can be expressed exactly and is not rounded, truncated or an unending decimal 

  • It may be written as an integer, a fraction, a terminating decimal or as a multiple of a root or known constant such as $\pi$

How do I give an answer in terms of π?

• In a question involving circles you may be asked to give an answer in terms of $\pi$

  • Treat $\pi$ like an algebraic term

• If calculating the circumference or area of a circle or part circle without a calculator

  • Substitute the values you know into the appropriate formula 

  • Simplify the expression by performing the operations on the numbers only

  • Leave your final answer as a multiple of $\pi$

Surds

What is a surd?

• A surd is the square root of a non-square integer

• Using surds lets you leave answers in exact form

  • E.g. $5\sqrt{2}$  rather than $7.071067812$

• Your calculator may automatically give you an answer to a calculation in surd form

  • Make sure that you are confident interpreting answers in surd form and entering surds into your calculator

How do I work with surds?

• Adding or subtracting surds is very like adding or subtracting letters in algebra

  • You can only add or subtract multiples of 'like' surds

    • $3\sqrt{5}+8\sqrt{5}=11\sqrt{5}$

    • $7\sqrt{3}–4\sqrt{3}=3\sqrt{3}$

• Be very careful, you can not add or subtract numbers under square roots

  • $\sqrt{9}+\sqrt{4}\ne \sqrt{9+4}$

    • $\sqrt{9}+\sqrt{4}=3+2=5$ 

    • $\sqrt{9+4}=\sqrt{13}=3.60555\dots$

• Remember that squaring is the inverse operation of taking the square root

  • If you square a surd, the result is the number without the root sign

  • ${\left(\sqrt{3}\right)}^2=\sqrt{3}\times \sqrt{3}=3$