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Using the fact that (✓a+✓b)²=a+b+2✓ab , or other- wise, determine the square root of 17 + 6√8 without using tables. Please help thank you!​

Sagot :

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Answer:

(a, b) = (9, 8) or (8, 9)

Step-by-step explanation:

Let the square of 17 + 6√8 be √a + √b that is,

[tex] \sqrt{17 + 6 \sqrt{8} } = \sqrt{a} + \sqrt{b} [/tex]

Square both sides

[tex]17 + 6 \sqrt{8} = {( \sqrt{a} + \sqrt{b} })^{2} [/tex]

Using the fact that (✓a + ✓b)² = a + b + 2✓ab

17 + 6√8 = a + b + 2√ab

This implies:

a + b = 17 ... (1) and

2√ab = 6√8 Divide both sides by 2

√ab = 3√8 Square both sides

ab = 9 × 8

ab = 72 ... (2)

From (1) b = 17 - a ... (3)

Substitute equation (3) in (2)

a ( 17 - a) = 72

17a - a² = 72

a² - 17a + 72 = 0 By Factorization

(a - 9)(a - 8) = 0

a = 9, a = 8

When a = 9

b = 17 - 9

b = 8

(a, b) = (9, 8)

When a = 8

b = 17 - 8

b = 9

(a, b) = (8, 9)

Therefore,

(a, b) = (9, 8) or (8, 9)

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