Laws on Integral Exponents

8/13/2019 Laws on Integral Exponents

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CHAPTER 1: Integral Exponents

I. Positive Integral Exponents Definition of a Positive-Integer Exponent

The expression xn is read as “x to the n th power” or simply “x to the n th ” , where x is called the b se

while n is called the exponent .

Laws of ExponentsA. Product of Power

Simply add the exponents if they have the same base.Examples: A) 2 3 24=2 3+4 =2 7

B) x3 x5=x 3+5 =X 8

B. Power of a Power

Simply multiply the exponents.Examples: A) (2 3) 4=2 (3)(4) =2 12

B) (y 4) 2=y (4)(2) =y 8

C. Power of a Product

Raise each factor inside the quantity to the given power or simply distribute the exponent insidethe quantity.Examples: A) (ab) 3=a 3 b3=a 3b3

B) (2x) 4=2 4 x4=16x 4

D. Power of Fraction

Raise all factors of both numerator and denominator to the given power.b 0 because any number divided by zero (0)will give an undefined answer.

Examples: A) ( )3 = , where y 0

B) ( ) 5 = =

Remember:

If n Z + (set of positive integers), then x n =x x x … x (n factors).

Remember:

For any numbera

R,

andm, n

Z +

:a m a n=a m+n

Remember:For any number a R, and m, n Z +:

(a m ) n=a mn

Remember:For any number a, b R, and m Z +:

(ab)m

=am

bm

Remember:

( ) m = ,

where m Z +; a, b R, and b 0.



E. Quotient of Powers

Simply subtract the exponents.

Take note that m>n or the exponent of the numerator is greater than the exponent of thedenominator and both are of the same base.

Examples: A) = 7 6-3 =7 3

B) = m 5-1 n4-1 =m 4n3

II. Zero and Negative Exponents

Definition of a Zero Exponent

Any number raised to the zeroth power, the answer is always 1 .Examples: A) (15a 6b3c8) 0=1

B) ( ) 0=1

Definition of a Negative Exponent

Bring down the negative exponents to make them positive.

Examples: A) a -5=

B) (3x) -3= =

Law of Negative Exponents

A. Reciprocal of a Negative Power

If the given is raised to a negative power, get the reciprocal of the numbers.a 0 because any number divided by zero (0)will give an undefined answer.

Examples: A) ( ) -5=( ) 5=

B) ( ) -2=( ) 2=

Remember:For all m, n Z + , where m>n ; and any a R

where a 0 : = a m-n

Remember:For any a R: a 0=1

Remember:For any a R and n Z -:

a -n =

Remember:

( ) -n = ,

where n Z -; a, b R, and a 0.



III. Evaluating Expressions with Zero and Negative Exponents

Summary of the laws of exponents for division:

Examples:

case 1: A) = 76-3 =7 3

B) ( ) = m 6-1 n4-1 =m 5n3

case 2: A) = x 10-10 =x 0=1

case 3: A)

Sol’n 1: = =

Sol’n 2: = r 2-8 =r -6=

IV. Simplifying Exponential ExpressionsExponential expressions are said to be in simplest form if:1. The exponents are positive,2. There are no powers of powers,

3.

Each base appears only once, and4. All fractions are in simplest form.

Examples:

A) Simplify where a, b 0.

Sol’n 1: = = 1 1 = 1

Sol’n 2: = = 1

B) Simplify

Sol’n 1: = = = 2 = =

C) Simplify 3x 2y0(18x -3y) -1

Sol’n 1: 3x2y0(18x -3y) -1 = 3(1)x 2 [ ] = 3x 2 ( ) = = =

D) Simplify

Sol’n 1: = = = = 1 = 1 = 1 1=1

Remember:For any number a R, where a 0 where and m, n Z +:

a m-n ; m>n (case1)

= a 0=1; m=n (case 2)

; m<n (case 3)

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Laws on Integral Exponents