Powers and roots appear in the quantitative section in forms that look unrelated: a large product that collapses to a single power, a comparison of two enormous numbers, and a fraction with a square root in the denominator. They are all the same handful of index laws applied in a different order.
Learn the laws once and a question such as which is larger, 3 to the power 30 or 2 to the power 45, becomes a two-line rewrite rather than a calculation.
Treat an exponent as a count of repeated multiplication, and three rules follow immediately.
The model assumes the base is a positive number. An even root of a negative number has no real value, and roots of negative numbers are outside the scope of the test.
The full set of index laws. The first three are the working rules; the rest extend them to zero, negative and fractional exponents.
| Law | Statement | Example |
|---|---|---|
| Product | am × an = am + n | 23 × 24 = 27 = 128 |
| Quotient | am ÷ an = am − n | 35 ÷ 32 = 33 = 27 |
| Power of a power | (am)n = amn | (52)3 = 56 = 15625 |
| Zero power | a0 = 1 | 70 = 1 |
| Negative power | a−n = 1 ÷ an | 2−3 = 1 ÷ 8 |
| Fractional power | a1/n = the n-th root of a | 91/2 = 3 |
Surds. A surd is a root that cannot be evaluated exactly, such as the square root of 2. The product rule for roots lets you pull out perfect squares.
Like surds add. root(50) plus root(18) simplifies to 5 root(2) plus 3 root(2), which is 8 root(2). Surds add only when the number under the root is the same.
Rationalising a denominator. Multiply top and bottom by the root in the denominator. For 6 divided by root(3), multiply both by root(3) to get 6 root(3) divided by 3, which is 2 root(3).
Comparing powers. Rewrite both sides with a common exponent or a common base. To compare 2 to the power 10 with 4 to the power 5, note that 4 is 2 squared and 4 to the power 5 becomes 2 to the power 10, so the two are equal.
Logarithms are the inverse of powers. logb x answers "b raised to what power gives x?". So log3 27 = 3 because 3³ = 27, and log2 32 = 5 because 25 = 32. Writing the definition once removes the mystery: blogb x = x and logb(bk) = k.
| Rule | Statement | Example |
|---|---|---|
| Product | logb(MN) = logb M + logb N | log2 8 + log2 4 = 3 + 2 = 5 = log2 32 |
| Quotient | logb(M/N) = logb M − logb N | log10 1000 − log10 10 = 3 − 1 = 2 |
| Power | logb(Mk) = k × logb M | log3 81 = log3 34 = 4 |
| Chain (telescoping) | loga b × logb c × ... = loga (last) | loga b × logb c × logc d = loga d |
The chain rule is change of base written as a product: each fraction log x / log (base) cancels against its neighbour, leaving only the first base and the last argument. A product that loops back on itself — loga n × logn a — is just 1.
Simplify root(50) plus root(18), and rationalise 6 divided by root(3).
Simplify each root by pulling out the largest perfect square. 50 is 25 times 2, so root(50) is 5 root(2). 18 is 9 times 2, so root(18) is 3 root(2).
For the second part, multiply numerator and denominator by root(3) to clear the root from the denominator.
Both answers are exact. A calculator gives root(50) plus root(18) as about 11.31, and 8 root(2) is also about 11.31, which confirms the simplification.
Logs, worked. log3 27 is the power that turns 3 into 27, which is 3. The product loga b × logb c × logc d × logd e × loge f telescopes: every intermediate base cancels, leaving loga f. A product that returns to its start — loga n × logn a — multiplies to 1 for the same reason.
Simplify 23 × 24.
Simplify the square root of 72.
Rationalise 4 divided by root(2).
Which is larger, 210 or 45?
What is log3 27?
Simplify loga b × logb c × logc d.
Without looking back: state the product, quotient and power-of-a-power laws, and the value of any nonzero number raised to the power zero.
The answers: same base multiply means add the exponents, same base divide means subtract the exponents, a power of a power means multiply the exponents, and any nonzero base to the power zero is 1.
For the same base, multiplying the powers means adding the exponents, so the new exponent is the of m and n. A power of a power the exponents. Any nonzero base raised to the power zero equals . The surds root(50) and root(18) simplify to 5 root(2) and 3 root(2), so their sum is 8 root(2), with 8 outside the root and inside it. log3 27 = because 3 cubed is 27. The chain loga b × logb c × logc d equals loga.
Which is larger, 330 or 245? Rewrite both to compare without a calculator.
Find a common exponent. Both 30 and 45 are multiples of 15, so write each power as a fifteenth power of a new base.
Both are raised to the same power, 15, so the larger base decides it. Since 9 is greater than 8, 3 to the power 30 is larger.
The alternative route, rewriting both with a common base, is dead here because 2 and 3 share no common base. The lesson is to look for a common exponent when the bases are unrelated, and a common base when one base is a power of the other.
Index laws are the grammar underneath algebra. The next lessons put them to work in equations and expressions, where powers move from being numbers to being variables, and where the same three rules reappear at every step.