Basic Concepts of Algebra

3. INDICES1, ROOTS AND RADICALS

3.  INDICES 1, ROOTS AND RADICALS

3.1 EXPONENTS:
                An exponent is the index/power to which a base is raised (see ch.2.2 #9). The expression x^5 which is read “x to the fifth power” has a base of x and an exponent /index/power 5. The index 5 in the expression  x^5, indicates how many times the base x is to be taken as a factor. Thus, x^5=x x x x x.

General form: If a is a real number and n is a positive integer, then a^n=aaaa…a                  n factors
Remember: a is the base and n is the index/exponents/power. n tells us how many times the base a is to be taken as a factor.
Examples:
1.  In the expression 5^2, the index 2 indicates that the number 5 is to be used as a factor twice.

Therefore, 5^2 = 5 x 5 = 25                      5 is used as a factor twice in its product.
     
2.  4x^2 means 4×x×x=4x^2            x^1  is used as a factor twice in its product.
                          (Only x is squared not 4)

3.  7^2 means 7 × 7 = 49
4.  〖12〗^2 means 12 × 12 = 144
5.  〖10〗^3 means 10 × 10 × 10 = 1,000
6.  (-5)^2 means -5 × -5 = +25
7.  (3a^2 )^3 means 3a^2  × 3a^2  × 3a^2=27a^(2+2+2)= 27a^6
             
8. (abc/2x)^2  means  abc/2x  × abc/2x=  (a^2 b^2 c^2)/(4x^2 )

Remember: An index can also be 1, 0, or a negative number. Any number with an index of 1 equals itself.  Any number (except 0) with an index of 0 equals 1.
Examples:
                  2^1=2          5^1=5          〖10〗^1=10
                  3^0=1          7^0=1          〖10〗^0=1   

To show that a^0=1,where a≠o. Consider the use of numerals:
                  5 x 3 = 15
                                               i.e. 5 = 5 .....corresponding items.
                15 ÷ 3 = 5
Also:
                a^0  x a^3=a^3     
                                            i.e. a^0=1 ......corresponding items.
                a^3  ÷a^3=1

     





      EXERCISE 3.1  Simplify:

1.  9^2

2.  (-4)^2=

3.  (-4a)^3=

4.  (a b^2 c^3 )^2=

5.  ((-3xy^2)/(4x^2 r^2 ))^2=

6.  7^0

7.  2^3∙2^3=

8. x^0∙x^0=

9.  (12x^3 )^°=

10. x^15/x^18 =






3.2 ROOTS AND RADICALS:
      We may recall that the inverse of multiplication is division and the inverse of addition is subtraction, vice versa. Similarly, the operation of raising a number to a power has its inverse, the operation is known as extracting the root of a number. Thus, the expression a^n is a base a raised to a power n. Given that a^n=m, Therefore, extracting the n^th root of m is indicated by nth √m) =a. The symbol √ is called the radical sign, m is termed the radicand, and n is known as the index.

For example:  Three factors of (2)              2x2x2=2^3=8,∴∛8=2
                          N factors of(a)                    axaxa…=a^n=m,∴ nth√(m)=a
Examples:
1.    √9 = 3 because  3^2=9  but 3^2=3 x 3 not 3 x 2
2.    √25=5 because  5^2=25

                    From examples 1 and 2 please note that the square root of a number or expression is one of its two equal factors. For example, +5 is a square root of 25 since (+5) (+5) = 25. Also, -5 is a square root of 25 since (-5) (-5) = 25. We may indicate the square root of 25 to be ± 5
                    The radical sign √ is used to indicate the principal square root. Thus √64 = +8. To indicate the negative square root of a number, a negative sign is placed before (on the left) of the symbol.
Thus, - √64 = -8
NOTE: By definition √0 = 0 →0^(1/2)  = 0
A radical is an indicated root of a number or expression such as √5, ∛27x, and ∜(11x^3 ).
The radicand is the number or expression under the radical sign. In the above example, the radicands are: 5, 27x, and 11x^3.
The index is a small number written above and to the left of the radical sign. The index indicates which root is to be taken. In square roots, the index 2 is not written.
Example:
                √12  means square root
                ∛27 means cube root
                ∜64 means fourth root







    Exercise 3.2    Simplify:

1.  ∛27 =

2.  5th√(32) =

3.  ∛125 =

4.  ∜81 =

5.  √(49/25) =

6.  ∛(27/x^6 ) =

7.  √(x^8/y^10 ) =

8.  ∜(1/16) =

9.  √(36/x^10 ) =

10.  -√(121a^6 ) =






3.3    SUMMARY EXERCISE for Ch3.  Simplify:
1.  √5 × √20

2.  √75

3.  √2a × √3b

4.  √0

5.  √3 × √6

6.  √(t^4 )

7.  √(25/x^2 )

8.  √20/5
9.  10√2 + 5 √2

10. √3x × √3x

11. √(12y^8 )

12. √(50x^4 )

13. √5 + ( √5 + √3  )

14. √49 + √64

15. ∜625

16. √(5&a^25 ) 

17. √(49/25)

18. √((4x^6)/(9x^4 ))

19. ∛(-1/8)

20.  5th√(32/243)