Practise › Questions › Indices and surds
Indices and surds questions
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.
7 original questions · 22 marks · the indices and surds notes · Algebra and functions
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
Evaluate 163/4, 50 and 4−2 without a calculator.
Worked answer
163/4 = (⁴√16)3 = 23 = 8. Then 50 = 1 and 4−2 = 1/16. B1 B1 B1, one for each value. Root before power keeps the numbers small; going the other way means cubing 16 first. A negative index means reciprocal, never a negative answer, and 4−2 = −16 is the slip this question exists to catch.Write √72 in the form k√2.
Worked answer
√72 = √(36 × 2) = √36 × √2 = 6√2. M1 for extracting a square factor, A1 for 6√2. Pull out the largest square factor in one step. Pulling out 4 first reaches the same place, but takes two rounds and gives the sign of a candidate who has not spotted the 36.Evaluate 8−2/3 and 253/2 without a calculator.
Worked answer
Work outside in. The minus means reciprocal and the 3 in the denominator means cube root, so 8−2/3 = 1/(∛8)2 = 1/4. Likewise 253/2 = (√25)3 = 53 = 125. M1 for treating the negative index as a reciprocal, A1 for 1/4, A1 for 125. The root and the power belong in the working separately; a bare answer from a calculator answers nothing on a non-calculator paper.Simplify √50 + √32 − √18.
Worked answer
Each surd reduces to a multiple of √2. √50 = 5√2, √32 = 4√2 and √18 = 3√2, so the sum is 5√2 + 4√2 − 3√2 = 6√2. M1 for reducing at least one surd, A1 for all three, A1 for 6√2. Like terms in √2 collect exactly as terms in x would. Adding the numbers under the roots instead gives √64 = 8, which is wrong and is a favourite distractor.Express 5/√5 and 4/√6 with rational denominators, in simplest form.
Worked answer
5/√5 = 5√5/5 = √5, since multiplying top and bottom by the surd clears it. And 4/√6 = 4√6/6 = 2√6/3. M1 for multiplying top and bottom by the surd, A1 for √5, A1 for 2√6/3. The final cancellation is part of the expected answer, so 4√6/6 left standing loses the mark.Express 10/(√6 − 2) in the form a√6 + b, where a and b are integers.
Worked answer
Multiply top and bottom by the conjugate √6 + 2. The denominator becomes 6 − 4 = 2 and the numerator becomes 10(√6 + 2), so the fraction is 10(√6 + 2)/2 = 5√6 + 10, giving a = 5 and b = 10. M1 for multiplying by the conjugate, A1 for the rational denominator, A1 for 5√6 + 10. The conjugate is chosen precisely so the cross terms cancel and the denominator turns rational. Multiplying by √6 − 2 instead leaves 10 − 4√6 downstairs, which is no better than where you started.Given that 2x = 8y+1 and 9y = 3x−9, find the value of x and the value of y.
Worked answer
Rewrite everything over a single base in each equation. The first gives 8y+1 = 23(y+1), so x = 3y + 3. The second gives 9y = 32y, so 2y = x − 9. Substituting the first into the second, 2y = 3y + 3 − 9, so y = 6 and then x = 21. Checking, 221 = 87 and 96 = 312, so both hold. M1 for rewriting the first equation over base 2, A1 for x = 3y + 3, M1 for rewriting the second over base 3, A1 for 2y = x − 9, A1 for y = 6 and x = 21. Equal bases forcing equal exponents is what converts an index problem into simultaneous linear equations, and each conversion carries a mark of its own. Logs would also work here, but they are slower and the numbers stop being exact.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise indices and surds one question at a time
The player marks nothing for you. It shows one question, waits, then shows the worked answer so you can mark yourself, and brings a question back sooner when it went badly.