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Grade 9:  The Learning Equation Math

11.07 Laws of Exponents 3

Numbers

Number Concepts

Refresher pp 14-15
 

SeekAWord2ech.class Author

Prerequisite Skills:

 

Key Terms                 

power, exponent, base, coefficient, simplify, evaluate

Interactive Component Under Construction

 

Learning Outcomes:

The student will:

 

Question Generator

The following will provide additional practice. Click on the "Simplify" button.  The solution will be given in a new window.   Return to this window and click on the "Random" button. See if you can determine the solution before clicking the "Simplify" button again.

  • ^ refers to an exponent.  3^3 refers to 32.
  • use / to represent divide and * to represent multiply.
  • use + and - for add and subtract.
  • any letter can be used as a variable.
    •

Help

Click to enter an expression, then click the button.   Try to calculate the answer mentally while the answer page loads.   The Power Property rules in the above chart may be useful.

 

Power and Power Property Question Generator

Your turn, simplify the following terms involving exponents. Your final answer should only involve, at most, positive exponents. To show an exponent, we shall use a ^, so 3^2 means 32 (which we would simplify to 9).  Click on the "Help" button if you need assistance to solve a question.

source:  http://www.polaris.nova.edu/MST/online/math1030/unit6/u6s1.html

Powers, Bases and Exponents:  Grade 9 Lesson 11.04

 

Review:

Explore the expanded and standard forms of powers with negative exponents:

Exponential Form (Power) Expanded Form/Repeated Multiplication Standard Form (Standard Name)

33

 3 x 3 x 3 27
32 3 x 3 9
31 3 3
30 no expanded form 1
3-1 = 1/31
1
3
 1/3
3-2 = 1/32
   1  
3 x 3
 1/9
3-3 = 1/33
      1     
3 x 3 x 3
 1/27
3-4 = 1/34
         1        
3 x 3 x 3 x 3
 1/81

Note:  The negative exponent indicates the reciprocal of the base for non-zero bases.

 

Power Property Example General Rule
addition 32 + 34 = 32 + 34

= 9 + 81 = 90

 xm + xn = xm + xn

Use BEDMAS to solve.
subtraction 32 - 34 = 32 - 34

= 9 - 81 = -72

 xm - xn = xm - xn

Use BEDMAS to solve.
product law 3334 = (3x3x3) (3x3x3x3)

= 33+4 = 37

 xmxn = xm+n

Keep the common base, then add the exponents.
x2x6 = (xx) (xxxxxx)

= x2+6 = x8
power of a power law (22)3 = (2x2)(2x2)(2x2)

= 22x3 =26

 (xm)n = xmn

Keep the base, then multiply  the exponents.
(x3)4 = (xxx)(xxx)(xxx)(xxx)

= x3x4 =312
power of a product law (3b)3 = (3b)(3b)(3b)

= (3x3x3)(BBB)

= 33b3

or

(3b)3 = (31b1)3 = 31x3b1x3 = 33b3

 (xy)m = xmym

If there are no exponents in the product, transfer the power to both factors.
(xy)4 = (xy)(xy)(xy)(xy)

= (xxxx)(eye)

= x4y4

or

(xy)4 = (x1y1)4 = x1x4y1x4 = x4y4
quotient law 35 ¸ 32 = (3x3x3x3x3)/(3x3)

= 35-2 = 33

 xm ¸ xn = xm-n

Keep the common base, then subtract the exponents.
x6 ¸ x2 = (xxxxxx)/(xx)

= x6-2 = x4
power of a quotient law (3/b)3 = (3/b)(3/b) (3/b)

= (3x3x3) /(bbb) = 27/b3

or

(3/b)3 = (31/b1)3 = 31x3/b1x3)

= 33/b3 = 27/b3

 (x/y)m = xm ¸ ym

If there are no exponents in the quotient, transfer the power to both parts of the fraction.
(x/y)4 =(x/y) (x/y) (x/y) (x/y)

= (xxxx) /(yyyy) = x4/y4

or

(x/y)4 = (x1/y1)4 = x1x4/y1x4 = x4/y4
negative exponents 3-4 = (1/3)4 = 1/34 x-m = (1/x)m = 1/xm

The negative exponents indicates the reciprocal of the base.  As a fraction

x = x/1

Thus the reciprocal of x is  1/x.
x-3 = (1/x)3 = 1/x3
1/y-4 = (y/1)4 = y4 1/x-m = xm

The negative exponents indicates the reciprocal of the base.   The reciprocal of  1/x  is x.

 

Exponent Laws

The following will provide additional practice. Click on the "Simplify" button.  The solution will be given in a new window.   Change the bases and/or exponents in this new window.  See if you can determine the solution before clicking the "Simplify" button again.

  • ^ refers to an exponent.  3^3 refers to 32.
  • use / to represent divide and * to represent multiply.
  • use + and - for add and subtract.
  • any letter can be used as a variable.

product law

xmxn = xm+n

power of a product law

(xy)m = xm  ym

power of a power law

(xm)n =xmn

quotient law

xm/ xn = xm-n

power of a quotient law

(x/y)m = xm ym

1/x-m = xm

 

Enrichment:

 

 

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