Surds & Indices

1Laws of indices, one base at a time

When two powers share a base, the four product-family laws collapse the expression: a^m \times a^n = a^{m+n}, \frac{a^m}{a^n} = a^{m-n}, (a^m)^n = a^{mn}, and a^0 = 1 for a \neq 0. The same laws applied after rewriting every base as a power of one prime — 8 = 2^3, 32 = 2^5 — are what solve almost every SSC indices equation without logarithms.

The trap is applying a law across different bases. 2^3 \times 3^3 is (2 \times 3)^3, not 6^6; the product-of-powers identity is (ab)^n = a^n b^n, which factors the base, never the exponent.

Figure. Same-base laws only fire after every number is a power of one prime. Here 32=2^5 turns the equation into 2^{x+3}=2^5, so the exponents equate and x=2. The trap the prose names — multiplying across different bases — never appears on this path because the rewrite forced one base first.

2Negative exponents are reciprocals

A negative exponent means take the reciprocal of the positive power: a^{-n} = \frac{1}{a^n} for a \neq 0. On a fraction the minus flips the fraction and then applies the positive power, so \left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2.

The product and quotient laws still hold with negative exponents — a^{-3} \times a^{5} = a^{2} — which is why rewriting first and then adding is safer than trying to picture a "negative power" as a smaller tower.

Figure. A negative exponent flips the fraction, then the positive power applies: (3/4)^{-2}=(4/3)^2=16/9. The side branch is the same claim written as a reciprocal of a positive power — both routes meet at 16/9.

How to clear a negative exponent

1. Flip, then powerWrite a^{-n} as \frac{1}{a^n}, or flip a fractional base and drop the minus.
2. Combine on one baseIf several powers of a remain, add the (signed) exponents before evaluating.
3. Evaluate the positive powerOnly after the sign is gone do you compute the ordinary integer power.

3Fractional exponents are roots

A fractional exponent is a root: a^{m/n} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}. The denominator is the order of the root; the numerator is the power. That identity is the bridge between indices and surds — every surd \sqrt[n]{a} is just a^{1/n} written with a radical sign.

For exam arithmetic, rewrite the base as a perfect n-th power first. 8^{2/3} becomes (2^{3})^{2/3} = 2^{2} = 4, which is faster than taking a cube root of 64.

Figure. Two legal routes to the same integer. The upper branch rewrites the base as a prime power and multiplies exponents; the lower takes the cube root first and then squares. Both land on 2^{2} = 4 — the fractional exponent is exactly that choice of order.

How to evaluate a^{m/n}

1. Prime-power the baseWrite a as p^{k} when you can, so the outer exponent can cancel.
2. Root, then powerCompute (\sqrt[n]{a})^{m} — the root usually lands on an integer before the power.
3. Or multiply exponentsOn a prime power, (p^{k})^{m/n} = p^{km/n} collapses in one step.

4Surds, and which ones may be added

A surd is an irrational root that cannot be simplified to a rational — \sqrt{2}, \sqrt[3]{5}, \sqrt{7}. Like surds share the same order and the same radicand, so their coefficients add exactly as like terms do: 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}. Unlike surds do not combine; \sqrt{2} + \sqrt{3} is already simplest.

Before adding, reduce each surd by factoring out a perfect square (or cube): \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}. The product and quotient rules \sqrt{a}\times\sqrt{b} = \sqrt{ab} and \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} are what make that reduction legal.

Figure. After reduction every term is a multiple of \sqrt{3}, so the figure is just the coefficients. 2 + 3 - 4 = 1 leaves a single \sqrt{3} — the radicand never changes when like surds combine.

How to combine surds

1. Reduce each radicandPull out every perfect-square factor so every term is k\sqrt{m} with m square-free.
2. Group like surdsOnly terms with identical m (and the same root order) may have their coefficients added.
3. Leave unlikes aloneDifferent radicands stay as a sum; do not invent a single root over their product.

5Rationalising a surd denominator

A surd in the denominator is cleared by multiplying numerator and denominator by a conjugate that produces a difference of squares. For a single term, \frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}. For a binomial, \frac{1}{\sqrt{a} + \sqrt{b}} \times \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}.

The conjugate flips the sign between the two surds; the denominator becomes the rational a - b. That is the whole method — the numerator just rides along.

Figure. Multiply top and bottom by the conjugate that flips the sign between the surds. The denominator becomes the rational difference of squares 5-3=2; the numerator \sqrt{5}+\sqrt{3} just rides along. That is the whole method.

How to rationalise

1. Name the conjugateFor \sqrt{a} \pm \sqrt{b}, the conjugate is \sqrt{a} \mp \sqrt{b} — flip the middle sign only.
2. Multiply top and bottomSame factor on numerator and denominator, so the value is unchanged.
3. Difference of squaresDenominator becomes a - b (or a for a monomial). Simplify the fraction if a common factor appears.

6Comparing surds of different orders

To compare \sqrt[m]{a} and \sqrt[n]{b}, raise both to the LCM of the orders so they share a common index, then compare the resulting radicands as ordinary integers. \sqrt{3} against \sqrt[3]{5} becomes a sixth-power comparison: 3^{3} = 27 against 5^{2} = 25, so \sqrt{3} is larger.

The common-index move is the same idea as rewriting bases before equating exponents — you are manufacturing a shared scale so the comparison is integer arithmetic, not a calculator estimate.

Figure. After both surds are raised to the sixth power, the comparison is 27 against 25 — ordinary integers. The dashed rule sits on 25 so the one-unit gap is readable; that gap is why \sqrt{3} wins.

How to compare

1. LCM of the ordersFor orders m and n, the common power is \mathrm{lcm}(m, n).
2. Raise each surd(\sqrt[m]{a})^{\mathrm{lcm}} = a^{\mathrm{lcm}/m}, an integer when a is integer.
3. Compare the integersThe larger raised value belongs to the larger surd — the common power is increasing on positives.