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Arithmetics with Exponents


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overview

Arithmetics with exponents, without evaluating the exponent, is explained. For example am×an=am+n. The list of formulas are derived using the first principles of exponent.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

first principles for all formulae

"base" repeatedly multiplied the "power" number of times equals "exponent".

The value of 25 is "25=2×2×2×2×2".

We learned the first principles of exponents as, for any numbers a and n,
an=a×a×a×(n times)

In this topic, the above is used to understand some known results.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

multiplication of exponents of same base

Consider 24×23?

Note that, by first principles
24=2×2×2×2
23=2×2×2

So, the given expression
24×23
=2×2×2×2×2×2×2
the simplified expression is 24+3=27

Generalizing this, for real numbers a, m and n,
am×an=am+n

multiplication of exponents of diff base

Consider 24×34?

Note : by first principles
24=2×2×2×2
34=3×3×3×3

So, the given expression
24×34
=2×2×2×2×3×3×3×3
=(2×3)×(2×3)×(2×3)×(2×3)
The simplified form of the expression is "(2×3)4"

Generalizing this, for real numbers a, m and n,
am×bm=(a×b)m

division of exponents of same base

Consider 24÷23

Note : by first principles
24=2×2×2×2
23=2×2×2

So, the given expression
24÷23
=2×2×2×2÷[2×2×2]
by properties of arithmetics
=2×2×2×2×[12×12×12]
The simplified expression is 24-3=21

Generalizing this, for real numbers a, m and n,
am÷an=am-n

division of exponents of diff base

Consider 24÷34

Note : by first principles
24=2×2×2×2
34=3×3×3×3

So, the given expression
24÷34
=2×2×2×2÷3×3×3×3
=2×2×2×2×13×13×13×13
=23×23×23×23
The simplified expression is (23)4

Generalizing this, for real numbers a, m and n,
am÷bm=(a÷b)m

exponents of power 0

Consider 30

Note that
30
=31-1
=31÷31
=33
The value of the expression is "1".

Generalizing this, for a real number a,
a0=1

exponents of exponent

Consider (52)3

Note that by first principles,
(52)3
=(52)×(52)×(52)
substituting 52=5×5
=(5×5)×(5×5)×(5×5)

The simplified expression is 52×3=56

Generalizing this, for real numbers a, m and n,
(am)n=amn

repeated addition of same exponents

consider 23+23+23+23+23

Note, by first principles a number repeatedly added is a multiplication, and so it is "5×23"

Generalizing this, for real numbers a, m and n,
am+am+(ntimes)=n×am

addition of multiples of exponents

Consider 3×25+4×25

Note that by first principles,
3×25=25+25+25
4×25=25+25+25+25

So,
3×25+4×25
=25+25+25+25+25+25+25

The simplified form of the above expression is "(3+4)×25=7×25".

Generalizing this, for real numbers a, m, p and q,
p×am+q×am=(p+q)×am

summary

Known results in Exponents :
For a,b,p,q,m,n,

am×an=am+n

am×bm=(a×b)m

am÷an=am-n

am÷bm=(a÷b)m

a0=1

(am)n=amn

a1m=am

am=1am

am+am+(n×)=n×am

p×am+q×am=(p+q)×am

Outline

The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

    →   Representation of Exponents

    →   Inverse of exponent : root

    →   Inverse of exponent : Logarithm

    →   Common and Natural Logarithms

    →   Exponents Arithmetics

    →   Logarithm Arithmetics

    →   Formulas

    →   Numerical Expressions

    →   PEMA / BOMA

    →   Squares and Square roots

    →   Cubes and Cube roots