THE EXPONENTS - EXPONENTIAL
FORM, EXPANDED FORM -
SOLVED EXAMPLES AND EXERCISE.
LINKS TO FURTHER STUDY

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Please study
BASICS OF THE EXPONENTS
before you study the examples given here.

Set of Solved Examples : Basics of The Exponents

Solved Example 1 : Basics of The Exponents

Write each of the following in exponential form.
(i) a.a.a (ii) 4.x.x.y.y (iii) a.a.b.b.b.c.c.c.c
(iv) 3.4.5.x.y.y.z.z.z
(v) 4(x + y)(x + y)(x + y)(x + y)
(vi) (a + 1) cubed
(vii) The product of 20 and the fourth power of (x - y)
(viii) The product of 25 and the fifth power of (a + b)

Solution:
Count the number of times each variable or term is multiplied
and write that number as exponent to the variable or term as base.

(i) a3 (ii) 4x2y2 (iii) a2b3c4 (iv) 60xy2z3
(v) 4(x + y)4 (vi) (a + 1)3(vii) 20(x - y)4(viii) 25(a + b)5

I hope you have followed the solution.

Now let us see another example which is the reverse of the above process.

Solved Example 2 : Basics of The Exponents

Solved Example 2 on Basics of The Exponents :

Write the term in the expanded form.
(i) 3x5 (ii) (4x)7 (iii) 4a3 (iv) (3xy)3 (v) 4(a - b)3

Solution:
Before expanding,
Identify the coefficient, base and the exponent in each term.
Keep the coefiicient aside
and multiply the base, exponent number of times.

(i) Here, coefficient = 3, base = x and exponent = 5.
Expanded form = 3.x.x.x.x.x
(ii) Here, coefficient = 1, base = 4x and exponent = 7.
Expanded form = 4x.4x.4x.4x.4x 4x.4x
(iii) Here, coefficient = 4, base = a and exponent = 3.
Expanded form = 4.a.a.a
(iv) Here, coefficient = 1, base = 3xy and exponent = 3.
Expanded form = 3xy.3xy.3xy
(v) Here, coefficient = 4, base = (a - b) and exponent = 3.
Expanded form = 4.(a - b).(a - b).(a - b)

To reinforce the idea of coefficient, base and exponent of a term,
let us see one more example.





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Solved Example 3 : Basics of The Exponents

Solved Example 3 on Basics of The Exponents :

Evaluate each expression for the given value of the variable:
(i) a7 when a = 1 (ii) (2a)5 when a = 5 (iii) 2x3 when x = 3
(iv) 3(2x - 1) when x = 9 (v) 3(x - 1)2 when x = 6
(vi) 4(x2 - 4) when x = 4 (vii) 4(x2 - 9) when x = -3

Solution:
Substitute the given value of the variable in the proper place.
Be careful about the base and coefficient.
In (ii), power 5 is to the base 2a,
where as in (iii), 2 is just coefficient and power 3 is to the base x.

(i) when a = 1, a7 = 17 = 1. Ans.
(ii) when a = 5, (2a)5 = (2x5)5 = 105 = 10x10x10x10x10 = 100000. Ans.
(iii) when x = 3, 2x3 = 2(3)3 = 2(3x3x3) = 2(27) = 54. Ans.
(iv) when x = 9, 3(2x - 1) = 3(2x9 - 1) = 3(18 - 1) = 3(17) = 51. Ans.
(v) when x = 6, 3(x - 1)2 = 3(6 - 1)2 = 3(5)2 = 3(5x5) = 3(25) = 75. Ans.
(vi) when x = 4, 4(x2 - 4) = 4(42 - 4) = 4(4x4 - 4)
= 4(16 - 4) = 4(12) = 48. Ans.
(vii) when x = -3, 4(x2 - 9) = 4{(-3)2 - 9} = 4{-3x-3 - 9}
= 4{9 - 9} = 4{0} = 0. Ans.

For further study of Exponents, go to

Laws of The Exponents,

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Exercise : Basics of The Exponents

Solve the following problems on The Exponents.

  1. Write each of the following in exponential form.
    (i) p x p x p x .....8 times
    (ii) 5.x.x.x.y.y.y.y
    (iii) 4(x - y)(x - y)(x - y)(x + y)(x + y)
    (iv) The product of 13 and fourth power of (x + y) (v) 32 (vi) 343
  2. Write the term in the expanded form.
    (i) 15x4y3z2
    (ii) (4xyz) (iii) 5(p + q)3 (iv) 53 x 52 (v) (45)3 (vi) 75⁄72
  3. Evaluate each set of expressions for the given value of the variable: (i) x3 and 3x when x = 5.
    (ii) 5x2 and (5x)2 when x = 2.
    (iii) 5 + x2 and (5 + x)2 when x = 3.
    (iv) x4 and 4x when x = 1.

For the Answers, see at the bottom of the page.

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Answers to Exercise : Basics of The Exponents

Answers to Problems on Basics of The Exponents :

  1. (i) p8
    (ii) 5x3y4
    (iii) 4(x - y)3(x + y)2 (iv) 13(x + y)4 (v) 25 (vi) 73
  2. (i) 15.x.x.x.x.y.y.y.z.z
    (ii) (4xyz).(4xyz).(4xyz).(4xyz)
    (iii) 5(p + q).(p + q).(p + q)
    (iv) 5 x 5 x 5 x 5 x 5
    (v) (4 x 4 x 4 x 4 x 4).(4 x 4 x 4 x 4 x 4).(4 x 4 x 4 x 4 x 4)
    (vi) (7 x 7 x 7 x 7 x 7)⁄(7 x 7)
  3. (i) 125 and 15
    (ii) 20 and 100
    (iii) 14 and 64
    (iv) 1 and 4.