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Indices and surds
Three laws govern every power there is, and once fractional and negative exponents join in, roots stop being a separate subject. Then surds, where the job is to keep an answer exact, tidy it down, and never let a root sit in a denominator.
IN THIS TOPIC
- Use the index laws for every rational exponent, negative and fractional included.
- Simplify a surd by its largest square factor and do surd arithmetic exactly.
- Rationalise a denominator, using the conjugate when the denominator has two terms.
COMMON MISCONCEPTION
√(a + b) = √a + √b.
The laws of indices
Everything about powers follows from three laws, and all three are on the must-learn list.
The course extends them to all rational exponents, and nothing about the extensions is a matter of taste. They are forced. Dividing a3 by a3 gives a0 by the second law and 1 by common sense, so a0 = 1. One more step and a−n = 1/an. Since (a1/2)2 = a1 by the third law, a1/2 has to be the square root of a, and in general am/n is the nth root of am. Write it whichever way suits the arithmetic.
Now the condition on the base, because two different claims get filed under one rule. For the laws to run over any real exponent the base must be positive, a > 0, and that is the line to quote. A single rational power is a smaller claim and survives a negative base whenever the exponent m/n, in lowest terms, has an odd n: (−8)1/3 = −2, since (−2)3 = −8, and (−32)3/5 = −8. An even n leaves nothing real, which is why (−4)1/2 does not exist.
Positivity is a condition on the laws, not on the values. Read (−8)1/3 as ((−8)2)1/6 and the third law gives 641/6 = 2, the wrong sign for a number whose cube must be −8. The value was real all along; the shuffling of exponents is what needed a > 0. So evaluate an odd-denominator power of a negative base directly, root first, and keep the index laws for positive bases.
WORKED EXAMPLE
A negative fractional power, unpacked
Evaluate 27−2/3 without a calculator.
Read it outside-in. The minus sign means reciprocal, the 3 underneath means cube root, the 2 on top means square.
27−2/3 = 1/272/3 = 1/(∛27)2 = 1/32 = 1/9.
Root first, power second. Squaring first would leave you cube-rooting 729. Same answer, worse afternoon.
Surds, kept exact
A surd is a root left unevaluated, √2 or 5√3, because writing 1.414… throws information away. Exact answers are the house currency of A-level maths. Surds inherit their rules from indices, and this is the one that does the work.
That, plus (√x)2 = x, and both want non-negative radicands, x ≥ 0 and y ≥ 0. A negative number has no real square root, so √(xy) = √x√y says nothing at all when x or y is negative, even where the left side exists: √((−4)(−9)) = 6, while neither root on the right is a real number. Read the rule right to left and it becomes the simplifying move, pulling the largest square factor out from under the root. Now notice what is missing from the list. Roots do not distribute over addition, and one counter example settles it in the style of the proof unit. √(9 + 16) = √25 = 5, while √9 + √16 = 7.
WORKED EXAMPLE
Simplifying a surd
Write √48 in the form k√3, and hence simplify √75 − √27.
√48 = √(16 × 3) = √16 × √3 = 4√3.
√75 = √(25 × 3) = 5√3 and √27 = √(9 × 3) = 3√3, so √75 − √27 = 2√3.
Terms in √3 collect the way terms in x would. Once each surd is fully simplified, surd arithmetic is ordinary algebra.
Rationalising the denominator
Convention wants denominators free of surds, and mark schemes follow convention. For a lone surd, multiply top and bottom by it, so 1/√2 becomes √2/2. When the denominator is a sum or a difference, multiply by its conjugate, the same expression with the middle sign flipped. The difference-of-squares identity (√x + √y)(√x − √y) = x − y then wipes the roots out.
WORKED EXAMPLE
The conjugate at work
Express 1/(3 − √2) with a rational denominator.
Multiply top and bottom by the conjugate 3 + √2. The denominator becomes (3 − √2)(3 + √2) = 9 − 2 = 7.
So 1/(3 − √2) = (3 + √2)/7.
Nothing else about the fraction changed, because (3 + √2)/(3 + √2) is 1. The conjugate is chosen precisely so that the cross terms cancel each other.
GUIDED PRACTICE
A fuller fraction
Express (2 + √5)/(3 − √5) in the form (a + b√5)/c, before opening the working.
Show the working
Multiply by (3 + √5)/(3 + √5). Denominator: 9 − 5 = 4.
Numerator: (2 + √5)(3 + √5) = 6 + 2√5 + 3√5 + 5 = 11 + 5√5. So the answer is (11 + 5√5)/4.
Expand the numerator like any other pair of brackets. Choosing the conjugate is the only step with any thinking in it.
INDEPENDENT PRACTICE
The quadratic you nearly missed
Solve, using algebra and showing each stage of your working, the equation x − 6√x + 4 = 0.
Show the working
Substitute u = √x, so u ≥ 0. The equation becomes u2 − 6u + 4 = 0.
By the quadratic formula, u = (6 ± √20)/2 = 3 ± √5. Both values are positive, so both survive.
Then x = u2 = (3 ± √5)2 = 9 ± 6√5 + 5 = 14 + 6√5 or 14 − 6√5.
Index laws to see the hidden quadratic, surd arithmetic to finish it, and a check that each value of u obeys u ≥ 0. Edexcel sets this equation almost word for word, so expect to meet it.
ASSESSMENT FOCUS
- The three index laws get quoted in working, never derived. Everything else, a0 = 1 and the negative and fractional powers, follows from them if a marker presses you.
- Take the root before the power. Writing am/n as (nth root of a)m keeps the numbers small, which is the entire point of a no-calculator question.
- State a > 0 when you apply the index laws with a general exponent. A rational power with an odd denominator, such as (−8)1/3 = −2, is still a real number, but it must be evaluated directly rather than shuffled through the laws.
- Simplify each surd fully before collecting anything. Pull out the largest square factor, then treat 4√3 the way you would treat 4x.
- "Show each stage of your working" means a method-mark scheme with no calculator behind it. Name your substitution, show the conjugate multiplication, and never leave a surd in a final denominator.
CHECK YOURSELF
Evaluate 323/5 without a calculator, and express 6/(√7 − 1) in the form a + √b.
Show a hint
Fifth root first; then the conjugate of √7 − 1 is √7 + 1.
Show the answer
323/5 = (⁵√32)3 = 23 = 8.
Multiply 6/(√7 − 1) by (√7 + 1)/(√7 + 1). The denominator becomes 7 − 1 = 6, so the fraction is 6(√7 + 1)/6 = √7 + 1, which is 1 + √7 in the requested form with a = 1 and b = 7.
Both answers are exact and neither needed a decimal at any stage. That is the standard this course holds answers to.
Three laws rule every power, and a fraction in the exponent is a root.
Take out the largest square factor; rationalise with the conjugate when the denominator has two terms.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the indices and surds questions page.
CHECK YOUR PROGRESS
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- Use the index laws for every rational exponent, negative and fractional included.
- Simplify a surd by its largest square factor and do surd arithmetic exactly.
- Rationalise a denominator, using the conjugate when the denominator has two terms.
Open the full revision checklist to see every objective in the course in one place.