Question

expand each expression using algebraic identities
1. (7-k^4)^2
2. (4x+y)^2​

Answer 1

Formula used -

(a - b)² = a² - 2ab + b²

(a + b)² = a² + 2ab + b²

1.

$(7 - {k}^{4} )^{2} \\ = ( {7)}^{2} - 2(7)( {k})^{4} + {( {k}^{4} )}^{2}$

[by the formula (a - b)² = a² - 2ab + b², here a = 7 and b = k⁴]

${7}^{2} - (2 \times 7 \times {k}^{4} ) + {( {k}^{4} )}^{2} \\ = 49 - 14 {k}^{4} + {k}^{8}$

Hence (7 - k⁴)² = 49 - 14k⁴ + k⁸

2.

(4x + y)²

= (4x)² + 2(4x)(y) + (y)²

[by the formula (a+b)² = a² + 2ab + b², here a = 4x and b = y]

= (4x)² + (2×4x×y) + y²

= 16x² + 8xy + y²

Hence (4x + y)² = 16x² + 8xy + y²

Some more formulas to remember, when bases are same (about powers) :-

${x}^{m} \times {x}^{n} = {x}^{m + n} \\ {x}^{m} \div {x}^{n} = {x}^{m - n} \\ ( {x}^{m} ) ^{n} = {x}^{m \times n} \\ (xy)^{m} = {x}^{m} {y}^{m}$

It only applicable when bases are same.

hope it helps.