​ Evaluate each of the following using identities: (i) (2x - 1/x)^2 (ii) (2x+y) (2x - y) (iii) (a^2b - b^2a)^2 ​

Question

Evaluate each of the following using identities:

(i) \((2x-\frac{1}{x})^2\)

(ii) (2x+y) (2x - y)

(iii) (a²b - b²a)²

(iv) (a – 0.1) (a +0.1)

(v) (1.5x² - 0.3y²)(1.5x² - 0.3y²)

Answer 1

(i) We know, (a - b)² = a² + b² -2ab

Here,

a = 2x and b = \(\frac{1}{x}\)

\((2x-\frac{1}{x})^2\) = \((2x)^2-(\frac{1}{x})^2 - 2\times x\times \frac{1}{x}\)

= \((4x)^2-(\frac{1}{x})^2 - 2\)

(ii) We know, (a-b)² = (a+b) (a - b)

(2x+y)(2x-y) = (2x)²– y² = 4x² – y²

(iii) We know, (a - b)² = a² + b² - 2ab

Here,

(a²b - b²a)² = (a²b)² + (b²a)² - 2 x a²b x b²a

= a⁴b² + a²b⁴ - 2a³b³

(iv) We know, (a-b)² = (a+b) (a - b)

(a – 0.1)(a +0.1) = (a)²– (0.1)²

= a² – 0.01

(v) We know, (a - b)² = a² - b²

(1.5x² - 0.3y²)(1.5x² - 0.3y²) = (1.5x²)² - (0.3y²)²

= 2.25x⁴ - 0.09y⁴