Powers, Indices and Roots: Rules and Estimation
Number • Structure and calculation
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Definition of power and index
A power expresses repeated multiplication of a base. The form a^n indicates the base a multiplied by itself n times when n is a positive integer. The index (exponent) n controls how many factors of a appear, and the base may be positive or negative depending on context. An index of 0 yields 1 for any non-zero base because multiplying zero times produces the multiplicative identity. A negative integer index produces a reciprocal effect, so a^(-n) equals 1 divided by a^n for non-zero a.
Basic laws of indices
The product rule applies when multiplying like bases: a^m × a^n equals a^(m+n) because exponents add when factors combine. The quotient rule applies when dividing like bases: a^m ÷ a^n equals a^(m−n) because common factors cancel. The power-of-a-power rule states (a^m)^n equals a^(m×n) because exponentiation repeats multiplication of exponents. These laws hold for all integer indices and extend to rational indices when definitions remain consistent. Limits appear when bases equal zero with negative or zero indices, and division by zero remains undefined.
Zero and negative indices
Any non-zero base raised to the zero index equals 1 because the product of zero factors defaults to the multiplicative identity. A negative integer index produces the reciprocal of the positive index: a^(-n) equals 1/a^n for a ≠ 0. Negative indices convert powers into fractions and therefore change the scale of a numerical value. Care with domain restrictions prevents undefined expressions. Zero raised to a non-positive index is undefined because division by zero or zero to the zero power lacks consistent meaning.
Fractional indices and roots
A fractional index a^(m/n) equals the n-th root of a^m and also equals (a^(1/n))^m by the power-of-a-power rule. The principal n-th root denotes the non-negative real root when n is even and when the radicand is non-negative. Fractional indices therefore convert between root notation and exponent notation to allow algebraic manipulation. Domain constraints appear for even denominators because the n-th root of a negative number is not a real number when n is even. Odd denominators allow negative radicands and produce a real root with sign preserved.
Recognising powers of small integers
Common powers of 2, 3, 4 and 5 form useful reference points for calculation and estimation. Familiarity with these small powers simplifies factorisation, calculation checks and estimation strategies. For example, powers of 2 grow as 2, 4, 8, 16, 32, 64; powers of 3 grow as 3, 9, 27, 81; powers of 4 grow as 4, 16, 64; powers of 5 grow as 5, 25, 125. Recognition of these values speeds mental arithmetic and supports bounding methods for non-integer powers and roots.
Estimating powers and roots
Estimation uses known powers as bounds. For a positive number x and integer n, the n-th root lies between two consecutive integer roots whose powers bound x. Logarithmic thinking or approximation via nearby powers produces a refined estimate when necessary. Rounding intermediate values and using a small number of arithmetic steps yields a practical approximate value. Estimation also uses inequalities: if a^n < x < b^n for integers a and b, then a < x^(1/n) < b. The same bounding idea applies to fractional indices by converting them into roots first.
Key notes
Important points to keep in mind