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

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

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.

Solved Example 3 : 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,

Exercise : Basics of 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.

Answers to Exercise : 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.