Find the values of each of the following:(i) 3-1 + 4-1(ii) (30 + 4-1)

1.	Find the values of each of the following:(i) 3-1 + 4-1(ii) (30 + 4-1) × 22(iii) (3-1 + 4-1 + 5-1)0(iv) ((1/3)-1 – (1/4)-1)-1
Answer» (i) 3^-1 + 4^-1 1/3 + 1/4 (we know that a^-n = 1/aⁿ) LCM of 3 and 4 is 12 = (1×4 + 1×3)/12 = (4+3)/12 = 7/12 (ii) (3⁰ + 4^-1) × 2² (1 + 1/4) × 4 (we know that a-n = 1/aⁿ, a⁰ = 1) LCM of 1 and 4 is 4 = (1×4 + 1×1)/4 × 4 = (4+1)/4 × 4 = 5/4 × 4 = 5 (iii) (3^-1 + 4^-1+ 5^-1)⁰ (We know that a⁰ = 1) (3^-1 + 4^-1 + 5^-1)⁰ = 1 (iv) ((1/3)^-1 – (1/4)^-1)^-1 (3¹ – 4¹)^-1 (we know that 1/a-n = an, a-n = 1/an) = (3-4)^-1 = (-1)^-1 = 1/-1 = -1

Find the values of each of the following:(i) 3-1 + 4-1(ii) (30 + 4-1) × 22(iii) (3-1 + 4-1 + 5-1)0(iv) ((1/3)-1 – (1/4)-1)-1

Answer»

(i) 3^-1 + 4^-1

1/3 + 1/4 (we know that a^-n = 1/aⁿ)

LCM of 3 and 4 is 12

= (1×4 + 1×3)/12

= (4+3)/12

= 7/12

(ii) (3⁰ + 4^-1) × 2²

(1 + 1/4) × 4 (we know that a-n = 1/aⁿ, a⁰ = 1)

LCM of 1 and 4 is 4

= (1×4 + 1×1)/4 × 4

= (4+1)/4 × 4

= 5/4 × 4

= 5

(iii) (3^-1 + 4^-1+ 5^-1)⁰

(We know that a⁰ = 1)

(3^-1 + 4^-1 + 5^-1)⁰ = 1

(iv) ((1/3)^-1 – (1/4)^-1)^-1

(3¹ – 4¹)^-1 (we know that 1/a-n = an, a-n = 1/an)

= (3-4)^-1

= (-1)^-1

= 1/-1

= -1

Simplify:(i) (4-1 × 3-1)2(ii) (5-1 ÷ 6-1)3(iii) (2-1 + 3-1)-1
Write each of the following in exponential form:(i) \((\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\)(ii) \((\frac{2}{5})^{-2}\times (\frac{2}{5})^{-2}\times (\frac{2}{5})^{-2}\)
By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1 ?
By what number should \((-15)^{-1}\) be divided so that the quotient may be equal to \((-5)^{-1}\)?
By what number should (1/2)-1 be multiplied so that the product may be equal to (-4/7)-1 ?
Find x, if(i) \((\frac{1}{4})^{-4}\times (\frac{1}{4})^{-8}=(\frac{1}{4})^{-4x}\)(ii) \((\frac{-1}{2})^{-19}\div (\frac{-1}{2})^{8}=(\frac{-1}{2})^{-2x+1}\)(iii) \((\frac{3}{2})^{-3}\times (\frac{3}{2})^{5}=(\frac{3}{2})^{-2x+1}\)(iv) \((\frac{2}{5})^{-3}\times (\frac{2}{5})^{15}=(\frac{2}{5})^{2+3x}\)(v) \((\frac{5}{4})^{-x}\div (\frac{5}{4})^{-4}=(\frac{5}{4})^{5}\)(vi) \((\frac{8}{3})^{2x+1}\times (\frac{8}{3})^{5}=(\frac{8}{3})^{x+2}\)
By what number should \(5^{-1}\) be multiplied so that the product may be equal to \((-7)^{-1}\) ?
By what number should 5-1 be multiplied so that the product may be equal to (-7)-1 ?
Find x, if(i) (1/4)-4 × (1/4)-8 = (1/4)-4x(ii) (-1/2)-19 ÷ (-1/2)8 = (-1/2)-2x+1(iii) (3/2)-3 × (3/2)5 = (3/2)2x+1
By what number should \((\frac{5}{3})^{-2}\) be multiplied so that the product may be \((\frac{7}{3})^{-1}\)?