Surds: Simplifying and Rationalising

The symbol √ gives the non-negative square root. For example, √36 = 6. A surd is an irrational root: √15 cannot be written as a fraction of two integers, so we keep the root symbol when we want its exact value.

To simplify √60, look for a square factor. Since 60 = 4 × 15, we can write √60 = √4 × √15 = 2√15. The remaining number 15 has no square factor greater than 1, so there is nothing more to take out.

Taking out a square factor changes the expression's form, not its value. An existing coefficient must stay in the calculation: 5√60 = 5 × 2√15 = 10√15.

A fraction can have a root in its denominator. Rationalising rewrites it as an equal fraction whose denominator is rational.

Consider 9/√5. Multiply by √5/√5. This factor equals 1, so the value stays the same: 9/√5 = (9 × √5)/(√5 × √5) = 9√5/5.

The denominator is now 5. The numerator still contains a root, which is allowed.

A numerical coefficient in the denominator stays in the product. For 7/(3√2), multiplying by √2/√2 gives 7√2/6 because 3√2 × √2 = 3 × 2 = 6. Multiply both numerator and denominator, then check whether their numerical coefficients have a common factor.

For addition, simplify the roots before comparing them. √32 + √98 becomes 4√2 + 7√2. Both terms contain √2, so their coefficients add to give 11√2. A sum such as √2 + √7 has different simplified root parts and stays as a sum.

For multiplication, multiply the numbers under the roots: √5 × √10 = √50 = 5√2. This product rule does not give a rule for adding radicands.

When a bracket is squared, write it twice and multiply every pair of terms: (√6 + 1)^2 = (√6 + 1)(√6 + 1) = 6 + √6 + √6 + 1 = 7 + 2√6.

In (√6 + 1)(√6 - 1), the middle products are -√6 and +√6. They cancel, leaving 6 - 1 = 5.

Simplifying and rationalising should preserve a number's value. Each equals sign says that the expressions on its two sides have the same value.

You can check a square-root simplification by squaring its proposed non-negative answer. For √147 = 7√3, the check is (7√3)^2 = 49 × 3 = 147. The proposed answer is non-negative, as the square-root symbol requires.

For rationalising, check the factor used. The step 11/√3 × √3/√3 uses a factor of 1 and gives 11√3/3. Multiplying only the denominator would use a different factor and change the value.

A rational denominator does not make the whole number rational. The expression 11√3/3 is still irrational. It is an exact form of the same number, not a rounded decimal.