Question

7. Simplify each of the following expressions:

(i) (3+√5)(2+√7) (ii) (5+√7)(5-√7) (iii) (√3 + √5)
2

(iv) (√7 − √3)
2​

Answer 1

Step-by-step explanation:

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Answer 2

Questions :

(i) (3+√5)(2+√7)

(ii) (5+√7)(5-√7)

(iii) (√3 + √5)²

(iv) (√7 − √3)²

Formula Used :

• (a + b)² = a² + 2ab + b²

• (a – b)² = a² – 2ab + b²

• (a + b)(a – b) = a² – b²

Solutions :

i) (3+√5)(2+√7)

→ 3(2 + √7) + √5(2 + √7)

→ 3•2 + 3•√7 + √5•2 + √5•√7

→ 6 + 3√7 + 2√5 + √35

ii) (5+√7)(5-√7)

Here we need to use the third formula in above i.e (a + b)(a – b) = a² – b²

→ (5)² – (√7)²

→ 5 × 5 – √7 × √7

→ 25 – 7

→ 18

iii) (√3 + √5)²

In this case we need to use the first formula i.e (a + b)² = a² + 2ab + b²

→ (√3)² + 2 × √3 × √5 + (√5)²

→ 3 + 2√6 + 5

→ 8 + 2√6

iv) (√7 − √3)²

Here the second formula will be used i.e (a – b)² = a² – 2ab + b²

→ (√7)² – 2•√7•√3 + (√3)²

→ 7 – 2√21 + 3

→ 10 – 2√21

Final Answers :

I) 6 + 3√7 + 2√5 + √35

II) 18

III) 8 + 2√6

IV) 10 – 2√21