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Arithmetic : Exponents, Powers and Roots of Numbers

 

Preparation Just what you need to know !

 

Roots

 

If an = c, then a = nc, 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 3c.
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, na = a1/n, √a = a1/2 and 3a = a1/3.
For example, 641/3 × 641/2 × 811/4 = 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 : a1/n x b1/n = (a b)1/n or n√(a b) = na x nb

 

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.


 

GMAT Math Review - Arithmetic : Index for Powers & Roots of Numbers
 

GMAT Math Review - Arithmetic : Practice Exercise for Powers & Roots of Numbers

 

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