Question

Simple :-

a) (5+√7)(2+√5)
b) (5+√5)(5-√5)
c) (√3+√7)²
d) (√11-√7)(11+√7)​

Answer 1

★ We know list some identities related to square roots, which are useful in various ways. The remaining ones follow from the distributive law of multiplication over addition of real numbers, and from the identity (x + y) (x - y) = x² - y², for any real numbers x and y.

Let a and b be positive real numbers. Then,

• √ab = √a √b

• √a/b = √a /√b

• (√a + √b) (√a - √b) = a - b

• (a + √b) (a - √b) ‍= a² - b

• (√a + √b) (√c + √d) = √ac + √ad + √bc + √bd

• (√a + √b)² = a + 2√ab + b

★ Qúestion (a)

(5 + √7) (2 + √5)

★ Solútion (a)

↦ (5 + √7) (2 + √5)

↦ 10 + 5√5 + 2√7 + √35

★ Qúestion (b)

(5 + √5) (5 - √5)

★ Solútion (b)

↦ (5 + √5) (5 - √5)

↦ 5² - (√5)² = 25 - 5 = 20

★ Qúestion (c)

(√3 + √7)²

★ Solútion (c)

↦ (√3 + √7)²

↦ (√3)² + 2√3√7 + (√7)²

↦ 3 + 2√21 + 7

↦ 10 + 2√21

★ Qúestion (d)

(√11 - √7) (11 + √7)

★ Solútion (d)

↦ (√11 - √7) (11 + √7)

↦ (√11)² - (√7)²

↦ 11 - 7

↦ 4

Answer 2

Answer :-

a)$\sf (5+ \sqrt{7})(2+\sqrt{5})$

$\sf = 5 ( 2 + \sqrt{5} ) + \sqrt{7} ( 2 + \sqrt{5} )$

$\sf = 10 + 5\sqrt{5} + 2\sqrt{7} + \sqrt{7} \times \sqrt{5}$

$\sf = 10 + 5\sqrt{5} + 2\sqrt{7} + \sqrt{35}$

$\boxed{\sf (5+ \sqrt{7})(2+\sqrt{5}) = 10 + 5\sqrt{5} + 2\sqrt{7} + \sqrt{35}}$

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b) $\sf (5+ \sqrt{5})(5- \sqrt{5})$

Using the identity -

• $\sf ( a + b ) ( a - b ) = a^2 - b^2$

$\sf = (5)^2 - (\sqrt{5})^2$

$\sf = 25 - 5$

$\sf = 20$

$\boxed{\sf (5+ \sqrt{5})(5- \sqrt{5}) = 20}$

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c) $\sf (\sqrt{3} + \sqrt{7})^2$

Using the identity -

• $\sf (a + b)^2 = a^2 + b^2 + 2ab$

$\sf = (\sqrt{3})^2 + (\sqrt{7})^2 + 2 ( \sqrt{3})(\sqrt{7})$

$\sf = 3 + 7 + 2 \sqrt{7 \times 3}$

$\sf = 10 + 2 \sqrt{21}$

$\boxed{\sf (\sqrt{3} + \sqrt{7})^2 = 10 + 2 \sqrt{21}}$

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d) $\sf (\sqrt{11} + \sqrt{7})(11- \sqrt{7})$

$\sf= \sqrt{11}(11- \sqrt{7}) + \sqrt{7}(11- \sqrt{7})$

$\sf= 11\sqrt{11} - \sqrt{11\times7} + 11\sqrt{7} - \sqrt{7\times 7}$

$\sf= 11 \sqrt{11} - \sqrt{77}+ 11\sqrt{7} - 7$

$\boxed{\sf (11+ \sqrt{7})(11- \sqrt{7}) = 11 \sqrt{11} - \sqrt{77}+ 11\sqrt{7} - 7 \:}$