Arithmetic : Exponents, Powers and Roots of Numbers

Preparation

Just what you need to know !

Roots

If a^{n} = c, then a = ^{n}√c, which is read as 'a is the nth root of c'.
The symbol √ is called the radical.
Thus, the nth root of a number c is a number that, when raised to the nth power, equals c.
For example, the 3rd root of 64 or ^{3}√64 = 4 because 4^{3} = 64, and the 6th root of 64 or ^{6}√64 = 2 because 2^{6} = 64.
The second root is called square root and is simply written (without the 2) as √.
The third root is called cube root and is written as ^{3}√.

Since 4^{2} = 4 × 4 = 16 and (−4)^{2} = (−4) × (−4) = 16, the square root of 16 is +4 and −4.
Thus, every positive number has two square roots, one positive and the other negative.
The symbol √16 denotes the positive square root, i.e., √16 = 4.
For example, the two square roots of 16 are √16 = 4 and −√16 = −4.

The square root of a negative number is not a real number because the the square of any number is non-negative.
For example, √−64 is not a real number, but ^{3}√−64 = −4 because (−4) × (−4) × (−4) = −64.

Every real number c has exactly one real cube root, which is denoted by ^{3}√c.
The cube root of a positive number is positive, and the cube root of a negative number is negative.
For example, ^{3}√125 = 5 because 5^{3} = 5 × 5 × 5 = 125 and ^{3}√−125 = −5 because (−5)^{3} = (−5) × (−5) × (−5) = −125.

Another way to write roots is as fractional exponents.
Thus, ^{n}√a = a^{1/n}, √a = a^{1/2} and ^{3}√a = a^{1/3}.
For example, 64^{1/3} × 64^{1/2} × 81^{1/4} = ^{3}√64 × √64 × ^{4}√81 = 4 × 8 × 3 = 96.

The second rule of exponents may be written for the case of roots as

MUST-KNOW : a^{1/n} x b^{1/n} = (a b)^{1/n} or ^{n}√(a b) = ^{n}√a x ^{n}√b

For example, √108 + √48 = √(36 × 3) + √(16 × 3) = (√36 × √3) + (√16 × √3) = 6 √3 + 4 √3 = 10 √3
10 √3 is called a radical expression or simply a radical.
Note that radicals can be added and subtracted only if the same number is under the radical sign.
For example, 6 √3 + 4 √3 = 10 √3, but 6 √2 + 4 √3 cannot be added and simplified.

Note that √2, √3, √5 and √7 are examples of irrational numbers.
Rational numbers (which include integers and fractions) and irrational numbers together form the set of real numbers that can be represented on the number line.