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Algebra and Functions

Indices and Surds

Pearson Edexcel International Advanced Level Mathematics

Laws of indices

LawStatement
Multiplyam × an = am + n
Divideam ÷ an = am − n
Power of a power(am)n = amn
Zero powera0 = 1
Negative powera−n = 1an
Fractional poweram/n = n√am = (n√a)m
  • A root becomes a fractional power: √x = x1/2, the nth root of x is x1/n.
  • A term in the denominator becomes a negative power: 1xn = x−n.
  • A power outside a bracket applies to the number as well as to x.

The form kxn

  • Give one term kxn, with k and n each a simplified constant.
  • Never write a power of x left in the denominator.
Method: write in the form kxn
  1. Write every root as a fractional power and every reciprocal as a negative power.
  2. Apply any power outside a bracket to the coefficient and to the power of x.
  3. Combine the coefficients into one number.
  4. Combine the powers of x with the index laws.
  5. Write one term kxn.
Trap: a root left in the answer

Never write a coefficient times a root of a power of x, such as a cube root or a square root left in the term. The power is not then a simplified constant. Write the root as a fractional power, so the answer is one coefficient times x to one fractional or negative power.

Worked example: express in the form kxn

Express each of the following in the form kxn, where k and n are simplified constants.
(a) 343√x   (b) (25x3)−1/2

(a) 3√x = x1/3, so 34x1/3 = 34x−1/3

34x−1/3

(b) 25−1/2 = 15 and (x3)−1/2 = x−3/2

15x−3/2

Index equations

  • Write both sides as powers of the same base.
  • When ap = aq, then p = q: equate the powers and solve.
Worked example: solve an index equation using a common base

In this question you must show all stages of your working.
Solve 4x + 1 = 8x√2

4x + 1 = 22x + 2

8x√2 = 23x × 2−1/2 = 23x − 1/2

2x + 2 = 3x − 12

x = 52

Surds

  • √a × √b = √ab and √a√b = ab
  • Simplify a surd by taking out the largest square factor: √m2n = m√n.
  • Rationalise 1√a by multiplying top and bottom by √a.
  • Rationalise 1a + √b or 1√a + √b by multiplying top and bottom by the conjugate, the same two terms with the sign between them changed.
  • Give surd answers in simplified surd form.
Method: rationalise a two-term denominator
  1. Multiply by the conjugate of the denominator over itself.
  2. Expand the numerator, term by term.
  3. Use the difference of two squares on the denominator, which leaves a whole number.
  4. Simplify each surd in the numerator and collect like surds.
  5. Divide each term by the denominator.
Worked example: rationalise to show a given form

Show that 5 + √23 − √2 can be written as 17 + 8√27

5 + √23 − √2 × 3 + √23 + √2

= 15 + 5√2 + 3√2 + 29 − 2

= 17 + 8√27

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