Arithmetic : Exponents, Powers and Roots of Numbers

Roots

If aⁿ = 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 4³ = 64, and the 6th root of 64 or ⁶√64 = 2 because 2⁶ = 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√.

The following square roots and cube roots are worth noting:
√1 = 1 ; √4 = 2 ; √9 = 3 ; √16 = 4 ; √25 = 5 ; √36 = 6 ; √49 = 7; √64 = 8; √81 = 9; √100 = 10 ; √121 = 11 ;
√144 = 12 ; √169 = 13 ; √196 = 14 ; √225 = 15 ; √256 = 16 ; √289 = 17 ; √324 = 18 ; √361 = 19 ; √400 = 20 ;
³√1 = 1 ; ³√8 = 2 ; ³√27 = 3; ³√64 = 4 ; ³√125 = 5 ; ³√216 = 6 ; ³√343 = 7 ; ³√512 = 8 ; ³√729 = 9 ; ³√1000 = 10 ;

Since 4² = 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 5³ = 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.

Example
Simplify 7 √80 − 6 √45 + 5 √63
Solution.

7 √80 − 6 √45 + 5 √63
= 7 √(16 × 5) − 6 √(9 × 5) + 5 √(9 × 7)
= 7 (√16 × √5) − 6 (√9 × √5) + 5 (√9 × √7)
= 7 × 4 √5 − 6 × 3 √5 + 5 × 3 √7
= 28 √5 − 18 √5 + 15 √7
= 10 √5 + 15 √7

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.