1.3.6: Exponents and Square Roots of Fractions

Learning Objectives

  • Simplify fractions raised to a power.
  • Simplify square roots of fractions.

Key words

  • Base: The number being raised to a power in an exponential expression
  • Exponent: The power in an exponential expression
  • Principal square root: The positive square root
  • Negative square root: The negative square root

Exponents

Exponents are used as a concise way to write multiple multiplication. For example, 2⋅2⋅2⋅2⋅2⋅2⋅2=272⋅2⋅2⋅2⋅2⋅2⋅2=27. The exponent 77 tells us how many times we have to multiply the base 22 by itself.

This same notation is used to raise a fraction to a power.

Examples

Identify the base and the exponent in each term:

1. (27)6(27)6                 Base = 2727 and exponent = 66

 

2.  (−59)4(−59)4               Base = −59−59 and exponent = 44

 

3.  (−125)3(−125)3             Base = −125−125 and exponent = 33

Try It

Identify the base and the exponent in each term:

1. (43)5(43)5                 Base = 4343 and exponent = 55

 

2.  (−18)7(−18)7               Base = −18−18 and exponent = 77

 

3.  (−115)3(−115)3             Base = −115−115 and exponent = 33

 

Examples

1. (23)2(23)2 = 23⋅23=4923⋅23=49

 

2. (−12)3(−12)3 = −12⋅−12⋅−12=−18−12⋅−12⋅−12=−18

 

3.  (−34)4(−34)4 = −34⋅−34⋅−34⋅−34=8164−34⋅−34⋅−34⋅−34=8164

 

4. (12)5=12⋅12⋅12⋅12⋅12=132(12)5=12⋅12⋅12⋅12⋅12=132

 

5. (−25)3=−25⋅−25⋅−25=−8125(−25)3=−25⋅−25⋅−25=−8125

 

6. (−27)2=−27⋅−27=449(−27)2=−27⋅−27=449

Try It

Simplify by writing the exponential term as multiple multiplication:

1. (45)2(45)2

 

2. (−25)3(−25)3

 

3. (−23)4(−23)4

 

Square Roots

The principal square root of a positive integer is the positive number that multiplies by itself to give the integer. As we saw in the section on integers, a positive integer has two square roots. The principal square root is positive and uses the radical sign for notation. For example, √4=2√4=2. The integer 44 is a perfect square since its square root is a whole number.  44 also has a negative square root notated with a negative sign in front of the radical: −√4=−2−√4=−2.

To find the square root of a fraction, we look for the fraction that squares to give the original fraction.

For example, to find the square root of 4949 we are looking for a fraction that when squared gives us 4949. Since 22=422=4 and 32=932=9, 2232=492232=49. Therefore, √49=23√49=23.

Notice that √4=2√4=2 and √9=3√9=3. This means that √49=√4√9=23√49=√4√9=23.

SQUARE ROOT RULE FOR FRACTIONS

√ab=√a√b√ab=√a√b for any whole numbers a,b;b≠0a,b;b≠0.

Examples

Simplify:

1. √2516=√25√16=54√2516=√25√16=54

 

2. −√814=−√81√4=−92

 

3. √121144=√121√144=1112

 

4. √−925=√−9√25 is undefined in the set of real numbers

Try It

Simplify:

1. √3649

 

2.  √10025

 

3.  −√16121

 

4.  √−1681