Question

If k = √(49 - 12√5), then find the value of k +2

Answer 1

Step-by-step explanation:

k = √(49 - 12√5)

= √(45 + 4 - 2 √180)

= √45 - √4

= 3 √5 - 2.

So, k + 2 = 3 √5 - 2 + 2 = 3 √5.

Answer 2

Answer:

k + 2 = 3√5

Explanation:

$\sf Given: k = \sqrt{49 - 12\sqrt{5} }$

$\sf \rightarrow k = \sqrt{49 - 12\sqrt{5} }$

step 1: breakdown

$\sf \rightarrow k = \sqrt{4 + 45- 2 \times 2 \times 3 \sqrt{5} }$

step 2: rewrite 45 = (3√5)²

$\sf \rightarrow k = \sqrt{4 + (3\sqrt{5})^2 - 2 \times 2 \times 3 \sqrt{5} }$

step 3: rearrange as per a² -2ab + b² = (a - b)²

$\sf \rightarrow k = \sqrt{ (3\sqrt{5})^2 - 2 \times 3 \sqrt{5} \times 2 +2^2}$

step 4: comparing, here: a = 3√5, b = 2

$\sf \rightarrow k = \sqrt{ (3\sqrt{5} -2)^2}$

step 5: simplify √a² = a

$\sf \rightarrow k = 3\sqrt{5} -2$

Then find k + 2

$\rightarrow \sf k = 3\sqrt{5} -2 + 2$

$\rightarrow \sf k = 3\sqrt{5}$