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# Arithmetic : Exponents, Powers and Roots of Numbers

 Preparation Just what you need to know !

Powers

If a is any number, then

a2 = a × a ;
a3 = a × a × a ; and
an = a × a × a × ... n times

where n is a positive integer.

a2 is read as 'a squared', a3 is read as 'a cubed', and an is read as 'a raised to the nth power'.
Thus, the nth power of a or an is the number a used n times as a factor in a product.
For example, 62 = 6 × 6 = 36, (−5)3 = (−5) × (−5) × (−5) = −125, and 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64.

The following squares and cubes are worth noting:
12 = 1 ; 22 = 4 ; 32 = 9 ; 42 = 16 ; 52 = 25 ; 62 = 36 ; 72 = 49 ; 82 = 64 ; 92 = 81 ; 102 = 100 ; 112 = 121 ;
122 = 144 ; 132 = 169 ; 142 = 196 ; 152 = 225 ; 162 = 256 ; 172 = 289 ; 182 = 324 ; 192 = 361 ; 202 = 400 ;
13 = 1 ; 23 = 8 ; 33 = 27 ; 43 = 64 ; 53 = 125 ; 63 = 216 ; 73 = 343 ; 83 = 512 ; 93 = 729 ; 103 = 1000 ;

Squaring a number that is greater than 1, or raising it to a higher power, yields a larger number.
For example, the powers of 3 are 32 = 3 × 3 = 9, 33 = 3 × 3 × 3 = 27, and 34 = 3 × 3 × 3 × 3 = 81.
Squaring a number that is between 0 and 1, or raising it to a higher power, yields a smaller number.
For example, the powers of (1/3) are (1/3)2 = 1/9, (1/3)3 = 1/27, and (1/3)4 = 1/81.
As another example, the powers of 0.1 are (0.1)2 = 0.01, (0.1)3 = 0.001, and (0.1)4 = 0.0001.

Note that 0n = 0 and 1n = 1 for any positive integer n.
a0 = 1 for any non-zero number a, and 00 is not defined.
a1 = a (i.e., if the power is 1, it is understood and usually not written).
a2 ≥ 0 for any a.
The square is always non-negative because the product of two negative numbers is positive.

If a fraction (a/b) is raised to the nth power, then (a/b)n = an/bn.

Example
If the value of an investment in the stock market increases by 25% each year, what will be the value of a \$8000 investment in 3 years?
Solution.

Each year, the investment increases by 25%, i.e., a factor of 1.25
In 3 years, the investment increases by a factor of 1.25 × 1.25 × 1.25 = (1.25)3
(1.25)3 = (5/4)3 = 53/43 = 125/64
\$8000 × 125/64 = 1000 × 125/8 = 1000 × 15.625 = 15625
The value of the investment in 3 years will be 15625.

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

10 more pages in GMAT Math Review

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