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.

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) 3x^{5} (ii) (4x)^{7} (iii) 4a^{3} (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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Evaluate each expression for the given value of the variable: (i) a^{7} when a = 1 (ii) (2a)^{5} when a = 5 (iii) 2x^{3} when x = 3 (iv) 3(2x - 1) when x = 9 (v) 3(x - 1)^{2} when x = 6 (vi) 4(x^{2} - 4) when x = 4 (vii) 4(x^{2} - 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, a^{7} = 1^{7} = 1. Ans. (ii) when a = 5, (2a)^{5} = (2x5)^{5} = 10^{5} = 10x10x10x10x10 = 100000. Ans. (iii) when x = 3, 2x^{3} = 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(x^{2} - 4) = 4(4^{2} - 4) = 4(4x4 - 4) = 4(16 - 4) = 4(12) = 48. Ans. (vii) when x = -3, 4(x^{2} - 9) = 4{(-3)^{2} - 9} = 4{-3x-3 - 9} = 4{9 - 9} = 4{0} = 0. Ans.

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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

Write the term in the expanded form. (i) 15x^{4}y^{3}z^{2} (ii) (4xyz) (iii) 5(p + q)^{3} (iv) 5^{3} x 5^{2} (v) (4^{5})^{3} (vi) 7^{5}⁄7^{2}

Evaluate each set of expressions for the given value of the variable: (i) x^{3} and 3x when x = 5. (ii) 5x^{2} and (5x)^{2} when x = 2. (iii) 5 + x^{2} and (5 + x)^{2} when x = 3. (iv) x^{4} and 4x when x = 1.

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(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)

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