Question

(3+√8)² + 1/(3+√8)²

please answer with steps and don't spawn
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Answer 1

Topic: Rationalization, Polynomials

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To Evaluate

$(3+\sqrt{8} )^2+\dfrac{1}{(3+\sqrt{8})^2}$

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Solution

What can we observe?

We see the two terms are all squares. If we try evaluating directly, the calculation will be long, since we should collect like terms.

We Can Observe

In this situation, we have some algebraic identities to use. Squaring both sides will double the exponent. Here, we are using $(a+b)^2=a^2+2ab+b^2$ on the sum.

Let $t=3+\sqrt{8}$. Then $\dfrac{1}{t} =3-\sqrt{8}$.

The value of $t+\dfrac{1}{t}$ is so 6.

$\rightarrow t+\dfrac{1}{t}=6$

Square both sides. Here, $a=t$ and $b=\dfrac{1}{t}$.

$\rightarrow (t+\dfrac{1}{t})^2=6^2$

$\rightarrow t^2+2\dfrac{t}{t} +\dfrac{1}{t^2} =36$

$\rightarrow t^2+\dfrac{1}{t^2} =34$

After we resubstitute the value of $t$ we get,

$\rightarrow (3+\sqrt{8} )^2+\dfrac{1}{(3+\sqrt{8} )^2} =34$

So, the value of evaluation is 34.

Answer 2

Identity used

In this problem, we are going to use the below identity:

$\rightarrow (a + b)^2 = a^2+b^2+2ab$

Solution:

Let us first add the following:

$\rightarrow 3+ \sqrt{8} + \dfrac{1}{3+ \sqrt{8}}$

On taking LCM, we get:

$\rightarrow \dfrac{(3+ \sqrt{8})(3+ \sqrt{8})+1}{3+ \sqrt{8}}$

On using identity, we get:

$\rightarrow \dfrac{9+8+6 \sqrt{8}+1}{3+ \sqrt{8}}$

$\rightarrow \dfrac{18+6 \sqrt{8}}{3+ \sqrt{8}}$

On taking 6 as common factor in numerator, we get:

$\rightarrow \dfrac{6(3+ \sqrt{8})}{3+ \sqrt{8}}$

$\rightarrow 6$

Therefore,

$\rightarrow 3+ \sqrt{8} + \dfrac{1}{3+ \sqrt{8}} = 6$

On squaring on both sides, we get:

$\rightarrow (3+ \sqrt{8} + \dfrac{1}{3+ \sqrt{8}})^2 = (6)^2$

Here,

• $a = 3+ \sqrt{8}$
• $b = \dfrac{1}{3+ \sqrt{8}}$

Again, by using identity, we get:

$\rightarrow (3+ \sqrt{8})^2 + (\dfrac{1}{3+ \sqrt{8}})^2+2 \times (3+ \sqrt{8}) \times (\dfrac{1}{3+ \sqrt{8}}) = 36$

$\rightarrow (3+ \sqrt{8})^2 + (\dfrac{1}{3+ \sqrt{8}})^2+2  = 36$

$\rightarrow (3+ \sqrt{8})^2 + (\dfrac{1}{3+ \sqrt{8}})^2 = 36-2$

$\rightarrow (3+ \sqrt{8})^2 + (\dfrac{1}{3+ \sqrt{8}})^2 = 34$