Lia's Math – Home page

Exponents (Powers) – pg.1/3

Posted in Math Theory by Lia on 06/28/2011

Exponents (Power): pg. 1, 23

Contents:

pg.1…..Exponents (Powers)  – about exponents (powers)

pg.2…..Exponents (Powers) – exercises

pg.3…..Exponents (Powers) – answers to exercises

 

About exponents (power)

An exponential expression has the form  {m^n} where {{}{n}{}} is called exponent and {{}{m}{}} is called base.

The exponential expression {m^n} can be read {{}{m}{}} to the power {{}{n}{}}.

The meaning of {m^n} is:

m^n = {underline {m*m*......... *m}under{multiply the number m,{}n times}}

Example:

{2^3} = {underline {2*2*2}under{multiply the number 2,{}3 times}} = 8

{(3/5)^4} = {underline {{3/5}*{3/5}*{3/5}*{3/5}}under{multiply the number {3/5},{}4 times}} =  {{81}/{625}}

or can be written:

{(3/5)^4} =  {{3^4}/{5^4}} = {{3*3*3*3}/{5*5*5*5}} = {{81}/{625}}

 

Exponential expressions properties.

1.  Addition and subtraction of exponential expressions.

Two exponential expressions cannot be added or subtracted unless they have same base and same exponent.

[1]  {m^n + m^p = m^n + m^p} but if  {p = n}, { m^n + m^n = 2 m^n}

[2]  {2m^n - m^p = 2m^n - m^p} but if  {p = n}, {2 m^n - m^n = m^n}

2. Multiplication of exponential expressions.

The result of multiplication between two exponential expressions with same base, is an exponential expression that has the same base as the factors’ base and the exponent, the sum of the factors’ exponents.

[3]

 {m^n} *{m^p} =  m^{n + p}

Example:  {4^5} *{4^2} = 4^{5 + 2} = 4^7

3. Division of exponential expressions.

The result of division between two  exponential expressions with same base, is an exponential expression that has the same base as the factors’ base and the exponent, the difference between the factors’  exponents.

[4]  

 {{m^n}/{m^p}} =  m^{n - p}

Example:  {{{7^5} /{7^2}} = {7^{5 - 2}} = {7^3}}

Note:

If  a division  has the dividend equal with 1 and the divider  an exponential expression, ( {{} {{m^p}}{}} ), we can consider that the dividend is <em>1</em> = {m^0}” title=”<em>1</em> = {m^0}”/> and proceed with the exponential expression division shown at <span style=[5]. 

{{1}/{m^p}} = {{m^0}/{m^p}} =  m^{0 - p} = m^{-p}

therefore: 

 {{1}/{m^p}} =  m^{-p}

 

Example:  {{1 /{6^3}} = {6^{- 3}}}

4. When the exponent is 0 the exponential expression is equal with 1 no matter what value the base has.

[5]

 {m^0} = 1

Example:  {2^0} = 1;{} {(3/7)^0} = 1;{}{}{{28}^0} = 1

5. When the base is 0 the exponential expression is equal with 0 no matter what value the exponent has.

[6]

 {0^n} = 0

Example:  {{0^3} = 0}; {0^{213}} = 0

6. When the exponent is 1 the exponential expression is equal with the value of the base. In this case the exponent is not written.

[7]  

  {m^1} = m

 

Example:  {{5^1} = 5}; {(2/5)^1} = {2/5}; {{32}^1} = 32

7. When the base is 1 the exponential expression is equal with 1 no matter what value the exponent has.

[8]  

  {1^n} = 1

Example:  {{1^3} = 1}; {1^{213}} = 1

8. When the exponent has the value 2, the exponential expression {{}{m^2}{}} is read m squared.

Example: {{23}^2} is read {{}{23}{}} squared.

9. When the exponent has the value 3, the exponential expression {m^3} is read m cubed.

Example:  {{4}^3} is read {{}{4}{}} cubed.

10. When multiplying two exponential expresions with different bases and same exponent the result is an exponential expression with same exponent and the base equal with the product between the factor’s bases.

[9]  

  {m^n}*{p^n} = (mp)^n

Example:  {2^5}*{3^5} = (2*3)^5= 6^5

or when there is a multiplication of two or more numbers raised to power the result can be written as shown bellow:

[10] 

  (mp)^n = {m^n}*{p^n}

Example:  6^5 = (2*3)^5 = {2^5}*{3^5}

11. When dividing two exponential expressions with different bases and same exponent the result is an exponential expression with same exponent and the base equal with the quotient  between the factor’s bases as shown.

[11] 

  {m^n}/{p^n} = (m/p)^n

Example:  {{8^5}/{2^5}} = (8/2)^5 = 4^5

or when there is a division of two numbers raised to power the result can be written as shown bellow:

[12] 

   {(m/p)^n} = {m^n}/{p^n}

Example:  (2/3)^5 = {{2^5}/{3^5}}

12. Raising to the power an exponential expression

The result of raising an exponential expression to the power is an exponential expression that has same base as the factor’s base and as exponent the product between the factor’s exponent and the power.

[13]  

{(m^n)^p} = {m^{np}}

Example:  {{(3^4)^5} = {3^{4*5}} = 3^20}

 

Exponents (Power): pg. 1, 23

 

 

 

 

 

 

Tagged with: ,

Leave a Reply