Powers, Roots & Laws of Indices

Chapter summary, hard words and model exam answers for ICSE Class 10 Hindi.

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Mathematics · CBSE 10 · ICSE 10 · GCSE (AQA, Edexcel, OCR)

In aⁿ the base a is multiplied by itself n times. So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. It is just shorthand for repeated multiplication.

Since 5² = 25, the square root √25 = 5. Since 3³ = 27, the cube root ∛27 = 3. A root is the reverse of a power.

These three rules let you simplify without multiplying everything out: 2³ × 2⁴ = 2⁷, 5⁶ ÷ 5² = 5⁴, (3²)⁴ = 3⁸.

a⁰ = 1 (from aⁿ ÷ aⁿ). A minus index means reciprocal: 2⁻³ = 1/8. A fraction index means a root: 8^(1/3) = ∛8 = 2.

base	the number being multiplied repeatedly in a power, the big number
index (exponent)	the small raised number showing how many times the base is multiplied
power	a base raised to an index, e.g. 2⁵
square root	the number that, multiplied by itself, gives the original; √
cube root	the number that, cubed, gives the original; ∛
reciprocal	one divided by the number; the reciprocal of a is 1/a
irrational number	a number whose decimal never ends or repeats, e.g. √2
fractional index	an index written as a fraction; denominator = root, numerator = power

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