Power and Roots

Why Powers and Roots Matter in CDS

A power (or exponent) is a short way of writing repeated multiplication. When we write 2⁵, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2 = 32. A root is the reverse operation: the fifth root of 32 brings us back to 2.

In the CDS Elementary Mathematics paper, this topic appears both directly (simplify 4^3/2, find a square root) and indirectly (inside geometry, surds in trigonometry, and number-system questions). Marks here are quick and almost guaranteed if your basics are solid.

Why does this matter so much under exam pressure? CDS gives you roughly ninety seconds per question. A candidate who hesitates over whether 16^3/4 means a root or a power burns half that time on a single step. A candidate with these laws automated reads the expression, converts the base to a prime power, and writes the answer in one clean line. Across a hundred-question paper, that fluency is the difference between attempting seventy questions and attempting ninety.

This chapter also builds the mental model you will reuse everywhere. Logarithms are simply the inverse of powers. Compound interest is repeated multiplication, which is exponentiation. The Pythagoras theorem produces square roots, and the diagonal of a cube produces √3 times a side. Treat Power and Roots as foundational rather than a small standalone topic.

The Laws of Indices

These eight rules govern every operation with powers. Memorise them so well that you apply them without thinking.

Notice that the product and quotient laws only work when the bases are the same. You cannot add the exponents of 2³ × 3² — the bases differ. When bases differ but exponents match, the other direction helps: a^mb^m can be collapsed to (ab)^m. So 2³ × 5³ = (2 × 5)³ = 10³ = 1000, which is far faster than computing 8 × 125 separately.

The fractional-exponent law is the single most useful rule for CDS, because it lets you treat every root as a power. Once a root becomes a power, all the other laws apply to it. There is no separate set of rules for roots — they obey exactly the same eight laws as integer powers. Internalising this saves you from memorising a second rulebook.

Zero, Negative and Fractional Powers

Three special cases trip up most candidates, so let us pin them down clearly.

Zero power

Any non-zero number raised to the power 0 equals 1. So 7⁰ = 1 and (−15)⁰ = 1. The value 0⁰ is left undefined and never appears in CDS.

Negative power

A negative exponent means take the reciprocal: 5⁻² = 1/5² = 1/25. Flipping a fraction turns the sign: (3/4)⁻² = (4/3)² = 16/9.

Fractional power

A fractional exponent combines a power and a root. For example 8^2/3 means the cube root of 8 squared: (8^1/3)² = 2² = 4.

Square Roots: Meaning and Methods

The square root of a number is the value that, when multiplied by itself, gives that number. The square root of 144 is 12 because 12 × 12 = 144. Symbolically √144 = 12.

Method 1: Prime factorisation

Break the number into prime factors, pair them, and take one factor from each pair. For √324: 324 = 2² × 3⁴, so √324 = 2 × 3² = 18.

Method 2: Long division

For large or non-perfect-square numbers, group digits in pairs from the right (the decimal point fixes the grouping), then find each digit of the root step by step. This is the reliable method for finding √1849 = 43 or estimating √2 ≈ 1.414.

Cube Roots and Higher Roots

The cube root of a number is the value whose cube gives that number. The cube root of 216 is 6 because 6³ = 216. We write ∛216 = 6.

Use prime factorisation in triples: 3375 = 3³ × 5³, so ∛3375 = 3 × 5 = 15. Unlike square roots, cube roots keep the sign of the number, so ∛(−27) = −3.

A handy shortcut for cube roots of perfect cubes: the last digit of the cube tells you the last digit of the root. A cube ending in 7 has a root ending in 3; a cube ending in 8 has a root ending in 2 — these pairs are mirror images apart from 0, 1, 4, 5, 6, 9 which map to themselves.

Surds and Their Order

A surd is an irrational root that cannot be simplified to a rational number, such as √2, √5 or ∛7. The number under the root is the radicand, and the small index n is the order of the surd.

A surd of the form √a is a quadratic (second-order) surd. To simplify a surd, pull out perfect-square factors: √72 = √(36 × 2) = 6√2.

Like and unlike surds

Surds with the same order and the same radicand are like surds and can be added or subtracted: 3√5 + 2√5 = 5√5. You cannot combine √5 and √3 by addition — they are unlike surds.

Rationalising the Denominator

Examiners love fractions with a surd in the denominator, because the standard move — rationalisation — tests whether you remember conjugates.

To rationalise 1/√3, multiply top and bottom by √3: 1/√3 = √3/3. To rationalise a denominator like (a + √b), multiply by its conjugate (a − √b), using the identity (x + y)(x − y) = x² − y².

This skill reappears in trigonometry and in algebra, so practise it until rationalising any denominator feels automatic. A typical follow-up question asks you to find the value of an expression such as (3 + √2)/(3 − √2) in the form a + b√2; you rationalise, expand, and read off a and b. Recognising that pattern in advance turns a two-minute problem into a forty-second one.

Worked Example: Combining the Laws

Let us solve a problem that blends indices, fractional powers and simplification — exactly the flavour CDS prefers.

Notice the strategy: rewrite every base as a power of a small prime, apply the power-of-a-power law, then do simple arithmetic. This three-step pattern cracks the majority of CDS index questions.

Worked Example: Rationalising a Conjugate

Here is a second solved illustration on the conjugate method, a frequent CDS pattern.

The answer (√5 + √3)/2 has a rational denominator, which is the required form. Always check that the denominator is now free of surds before you stop.

Common Mistakes to Avoid

Most lost marks in this topic come from a handful of repeated slips. Train yourself to catch them.

• Adding exponents when bases differ, e.g. 2³ × 3² ≠ 6⁵.
• Treating a negative exponent as a negative answer.
• Writing √(a+b) = √a + √b.
• Forgetting that √x² = |x|, so √((−4)²) = 4, not −4.
• Leaving a surd in the denominator when the question asks for a rationalised form.

Previous-Year Style Practice

Test the full toolkit on a question in the genuine CDS pattern.

The lesson: when several powers are equal, set them all to a common value and convert the exponents into one variable. This collapses a scary-looking problem into simple algebra.

Quick Revision

Run through this checklist the night before your exam.

Drill ten mixed problems daily from your PYQ book for a week, and Power and Roots will become one of your fastest-scoring CDS topics.