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

Sagot :

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.

fieryanswererft

Answer:

k + 2 = 3√5

Explanation:

[tex]\sf Given: k = \sqrt{49 - 12\sqrt{5} }[/tex]

[tex]\sf \rightarrow k = \sqrt{49 - 12\sqrt{5} }[/tex]

step 1: breakdown

[tex]\sf \rightarrow k = \sqrt{4 + 45- 2 \times 2 \times 3 \sqrt{5} }[/tex]

step 2: rewrite 45 = (3√5)²

[tex]\sf \rightarrow k = \sqrt{4 + (3\sqrt{5})^2 - 2 \times 2 \times 3 \sqrt{5} }[/tex]

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

[tex]\sf \rightarrow k = \sqrt{ (3\sqrt{5})^2 - 2 \times 3 \sqrt{5} \times 2 +2^2}[/tex]

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

[tex]\sf \rightarrow k = \sqrt{ (3\sqrt{5} -2)^2}[/tex]

step 5: simplify √a² = a

[tex]\sf \rightarrow k = 3\sqrt{5} -2[/tex]

Then find k + 2

[tex]\rightarrow \sf k = 3\sqrt{5} -2 + 2[/tex]

[tex]\rightarrow \sf k = 3\sqrt{5}[/tex]

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