Sets, surds, indices and logarithms

What you need to know

• A set is a well-defined collection of objects, written either by listing, {2, 3, 5, 7}, or by rule, {x : x is prime and x < 10}. A set with n distinct elements has 2^n subsets, so {a, b, c} has 2^3 = 8 subsets, counting the empty set and the set itself.
• For two sets use n(A u B) = n(A) + n(B) - n(A n B). When a question gives you totals and asks for the overlap, draw the Venn diagram and fill the intersection first, then subtract outwards, so that nobody is counted twice.
• For three sets, fill the central region n(A n B n C) first, then each pairwise region minus the centre, then each single region. Every region must carry a number before you write an answer, and all the regions plus the outside must add up to the universal set.
• De Morgan's laws say (A u B)' = A' n B' and (A n B)' = A' u B'. The complement A' means everything in the universal set that is not in A, so questions phrased as neither or not both are usually complement questions in disguise.
• A surd is an irrational root left in root form. Always pull out the largest perfect square: sqrt(50) = sqrt(25 x 2) = 5 sqrt(2) and sqrt(72) = sqrt(36 x 2) = 6 sqrt(2). Leaving sqrt(50) as it is will cost you the simplification mark.
• You may add or subtract surds only when the surd part is identical. sqrt(8) + sqrt(18) looks impossible until you simplify each one: 2 sqrt(2) + 3 sqrt(2) = 5 sqrt(2). But 3 sqrt(2) + 5 sqrt(3) stays exactly as it is.
• To rationalise a single surd denominator, multiply top and bottom by that surd: 3/sqrt(5) becomes 3 sqrt(5)/5. For a denominator of the form a + b sqrt(c), multiply by the conjugate a - b sqrt(c), because the denominator then becomes a^2 - b^2 c, which is rational.
• The index laws are a^m x a^n = a^(m+n), a^m / a^n = a^(m-n) and (a^m)^n = a^(mn). Anything to the power zero is 1, and a^(-n) = 1/a^n. Negative powers mean reciprocal, not negative answers.
• In a^(m/n) the denominator n is the root and the numerator m is the power. So 8^(2/3) is the cube root of 8, squared, which is 2^2 = 4, and 16^(-3/4) = 1/(16^(3/4)) = 1/2^3 = 1/8.
• Any equation containing both a^(2x) and a^x is a hidden quadratic. Substitute y = a^x, solve the quadratic in y, then convert each value of y back to x. Reject any negative value of y straight away, because a^x can never be negative.
• log_a N = x and a^x = N say exactly the same thing. Most logarithm questions are solved by switching between the two forms, so log_2 32 = 5 simply because 2^5 = 32. Remember that log_a a = 1 and log_a 1 = 0.
• The log laws are log(MN) = log M + log N, log(M/N) = log M - log N and log(M^p) = p log M. Change of base is log_a N = (log_b N)/(log_b a), which is how you handle a question mixing log_2 and log_8.
• After solving a logarithmic equation, substitute every root back into the original expression. Any root that makes an argument zero or negative must be rejected, because the logarithm of a non-positive number does not exist.