Question

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Answer 1

Question:-

Simplify the following using identities.

a) (109)² + (91)²

b) (200)² - (100)²

c) (1025)² - (975)²

d) (10)² + (20)²

Solutions:-

a) (109)² + (91)²

$\sf{\longrightarrow}$ Let us break the expressions such that they come in the form (a+b)² and (a-b)²

So,

(109)² = (100 + 9)²

(91)² = (100-9)²

Therefore,

[(100+9)² + (100-9)²]

Applying,

(a+b)² = a² + b² + 2ab

(a+b)² = a² + b² + 2ab(a-b)² = a² + b² - 2ab

Here a = 100 and b = 9

= $\sf{[(100)^2 + (9)^2 + 2\times100\times9] + [(100)^2 + (9)^2 - 2\times100\times9]}$

= $\sf{[10000 + 81 + 1800] + [10000 + 81 - 1800]}$

= $\sf{10000 + 81 + \cancel{1800} + 10000 + 81 - \cancel{1800}}$

= $\sf{10081 + 10081}$

= $\sf{10162}$

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b) (200)² - (100)²

$\sf{\longrightarrow}$ Applying, a² - b² = (a+b)(a-b)

Here a = 200 and b = 100

$\sf{(200+100)(200-100)}$

= $\sf{300\times100}$

= $\sf{30000}$

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c) (1025)² - (975)²

$\sf{\longrightarrow}$Applying, a² - b² = (a+b)(a-b)

Here a = 1025 and b = 975

$\sf{(1025+975)(1025-975)}$

= $\sf{2000\times50}$

= $\sf{100000}$

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d) (10)² + (20)²

$\sf{\longrightarrow}$ Let us break the expressions such that they come in the form of (a+b)² and (a-b)²

So,

(10)² = (10+0)²

(20)² = (10+10)²

Therefore,

Applying,

(a+b)² = a² + b² + 2ab

Here,

For (10)² a = 10 and b = 0

And For (20)² a = 10 and b = 10

= $\sf{[(10)² + (0)² + 2\times0\times10] + [(10)² + (10)² + 2\times10\times10]}$

= $\sf{(100 + 0 + 0) + (100 + 100 + 200)}$

= $\sf{100 + 400}$

= $\sf{500}$

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Additional Information:-

• $\sf{(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca)}$

• $\sf{(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}$

• $\sf{(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3}$

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