Machinery's Handbook, 31st Edition
PERCENTAGE 11 previous example, production increased from 280 to 300, then the percent change would be: final original amount original 300 280 − × = − × = 100 280 100 20 280 ×100 714 = . % . The denominator is always the original amount. Percent change also can be neg- ative. If production decreased from 280 to 245, the percent change would be: 245 280 280 100 35 280 100 125 − × = − × = − . % . Powers and Roots Powers: The square or second power of a number (or quantity) is the product of that number multiplied by itself. Thus, the square of 9 is 9 × 9. The square of a number is indi- cated by the exponent 2 , thus: 9 2 = 9 × 9 = 81. The cube or third power of a number n is the product n × n × n , or n 3 . Thus, the cube of 4 is 4 × 4 × 4 = 64, and is written 4 3 . In general, the n th power of a is written a n , where a is the base and n is the exponent . Roots: The square root of a given number is the positive number which, when multiplied by itself, will produce the given number. The square root of 16 (written 16 ) is 4 because 4 × 4 = 16. The other root of 16 is –4, but the use of the square root symbol indicates the positive (principal) square root only. Similarly, the cube root of a given number is the number which, when used as a factor three times, will produce the given number. Thus, the cube root of 64 (written 64 3 ) is 4 because 4 × 4 × 4 = 64. In general, the n th root of a is written as a n or a 1/ n . Properties of Exponents a n a m a n m + = a n a m --- a n m – ( ) = a m ( ) n a mn = ab ( ) m a m b m = a n – 1 a n = --- 1 a n – ---- a n = a 0 1 a 0 ( ≠ ) = a 1 n / a n = a b n n = ( ) a b n a m n / a 1 n / ( ) m = or ( a m ) 1/ n a m n / a n ( ) m = = unless a < 0, and m and n are both even
a n m ( )
Examples: 3 1 3 2
x 1+3 ( )
3 1+2
3 3 = =
x ( ) x 3 ( )
x 4
=
27
=
=
5 4 5 2 --- 5 4 – 2 =
x 9 x 6 ---
x 9 – 6 ( )
5 2 = =
x 3
25
=
=
2 4 ( ) 2 ( )
x 3 ( ) 3 ( )
2 4 ( ) 2
x 3 ( ) 3
2 8 = =
x 9
=
256
=
=
9 x ( ) 2
ab 4 ( ) 2
9 2 x 2
81 x 2
a 2 b 8 =
=
=
3
1 4 3 --- 1 64 --- = =
32 5 ( )
32 1/5 ( ) 3
32 3/5
=
=
4 –3
= 2 3 = 8
1 2 –5 ---- 2 5
32 = =
9 1 ( )
9 x 0
=
=
9
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