Surds and Indices: Formulas, Rules and Solved Examples
Surds and Indices covers irrational roots and powers written as exponents. This topic uses laws of indices, fractional exponents, surd operations and rationalisation. Learn how to simplify expressions such as √72, 16^(3/4), and 1/(2 + √3) using standard formulas and exact calculations.
What Are Surds and Indices?
A radical such as √[n]{a} can be written as a fractional index: √[n]{a} = a^(1/n). More generally, a^(m/n) = √[n]{a^m}. Surds are usually simplified by taking perfect powers outside the radical. Thus, √72 = √(36 × 2) = 6√2.
Surds and Indices Formula & Tricks
Important Formulas
When powers have the same non-zero base, add their indices.
When powers have the same base, subtract the denominator index from the numerator index.
Multiply the indices when one power is raised to another power.
The denominator gives the root and the numerator gives the power.
For real values, a and b must be non-negative, and b must be positive in the quotient.
Multiply the numerator and denominator by √a to remove the radical from the denominator.
Quick Tricks
Rewrite radicals as powers before applying index laws. This reduces mixed root-and-power expressions to one form.
Factor the number under a radical into the largest perfect square, cube or nth power available.
For a denominator containing a sum or difference with a square root, multiply by its conjugate. The denominator then becomes a difference of squares.
Surds and Indices Concepts
Laws of Indices
For a non-zero base, a^0 = 1, a^(−n) = 1/a^n, a^m × a^n = a^(m+n), and a^m/a^n = a^(m−n). Also, (ab)^n = a^n b^n and (a/b)^n = a^n/b^n when the expressions are defined. These rules must not be applied by adding or multiplying bases.
Fractional and Negative Indices
The expression a^(1/n) means the nth root of a, while a^(m/n) means the nth root of a^m. For a ≠ 0, a^(−n) = 1/a^n. In real-number calculations, an even root requires a non-negative radicand.
Simplification and Operations on Surds
Like surds have the same irrational part, such as 2√3 and 5√3. Their coefficients can be combined, but unlike surds cannot be directly added. Multiplication and division use radical product and quotient rules.
Rationalisation of Surd Denominators
For a denominator √a, multiply by √a. For a denominator a + √b or a − √b, multiply by the conjugate a − √b or a + √b. The identity (a + √b)(a − √b) = a^2 − b produces a rational denominator when a^2 − b is rational and non-zero.
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Surds and Indices Quick Revision
Use these rules to simplify powers, roots and rational expressions.
- a^m × a^n = a^(m+n) and a^m/a^n = a^(m−n), for a ≠ 0.
- (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n.
- a^(m/n) = √[n]{a^m}; the denominator is the root and the numerator is the power.
- Extract perfect powers from radicals: √72 = 6√2 and ∛54 = 3∛2.
- Only like surds can be added or subtracted.
- Use a surd itself to rationalise a single-root denominator and use the conjugate for a binomial denominator.
- For real values, even roots have non-negative radicands.
Surds and Indices FAQs
What is the value of 27^(2/3)?
27^(2/3) = (∛27)^2 = 3^2 = 9.
How is √75 simplified?
√75 = √(25 × 3) = 5√3.
Can √2 and √8 be added directly?
First simplify √8 = 2√2. Therefore, √2 + √8 = √2 + 2√2 = 3√2.
What is the rationalised form of 1/(3 − √2)?
Multiply by the conjugate: 1/(3 − √2) = (3 + √2)/(9 − 2) = (3 + √2)/7.
What is the difference between a surd and an index?
A surd is an irrational root such as √5. An index is a power or exponent, such as 3 in 2^3.
Why is a^0 equal to 1?
Using a^m/a^n = a^(m−n), a^n/a^n = a^(n−n) = a^0. Since a^n/a^n = 1 for a ≠ 0, a^0 = 1.