Question

If a = (3-2√2) find the value of a4+ 1/a4 ​

Answer 1

Answer:

  1154

Step-by-step explanation:

Perhaps the easiest way to evaluate this numerical expression is to let a calculator do it. Alternatively, we can compute the value from a +1/a.

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expression

Consider the square of a +1/a:

  (a +1/a)^2 = a^2 +2(a)(1/a) +1/a^2 = (a^2 +1/a^2) +2

This means ...

  a^2 +1/a^2 = (a +1/a)^2 -2

Similarly, using a^2 for 'a' in the above, we have ...

  a^4 +1/a^4 = (a^2 +1/a^2)^2 -2

numerical value

The value of a +1/a is ...

  $a+\dfrac{1}{a}=3-2\sqrt{2}+\dfrac{1}{3-2\sqrt{2}}=(3-2\sqrt{2})+\dfrac{3+2\sqrt{2}}{3^2-(2\sqrt{2})^2}\\\\=(3-2\sqrt{2})+(3+2\sqrt{2})\\\\a+\dfrac{1}{a}=6$

Then the value of a^2 +1/a^2 is 6^2 -2 = 34

and the value of a^4 +1/a^4 is 34^2 -2 = 1154