Question

17` IF x = 3 + 2√2, Then x + 1/x is a​

Answer 1

Step-by-step explanation:

Let √x - 1/√x = a

Squaring both the sides,

x + 1/x - 2 = a^2

Putting the value,

3–2√2 + 1/(3–2√2) - 2 = a^2

a^2 = 1 - 2√2 + 1/(3–2√2)

= [(3–2√2) (1–2√2) + 1] / 3–2√2

= {3 - 8√2 + 9} / 3–2√2

= [12 - 8√2] / 3–2√2

Rationalising both the sides

= {(12 - 8√2)(3+2√2)} ÷ (9–8)

= 36 + 24√2 - 24√2 + 16(2)

= 36 - 32

=> 4

a^2 = 4

a = √4

a = 2, -2

So,

√x - 1/√x = a = 2, -2.

3–2√2 = 1–2√2+2= 1^2–2×1×√2+(√2)^2= (1-√2)^2

x=(1-√2)^2 , √x=1-√2

√x-1/√x=1-√2–1/(1-√2)=(1-√2-√2+2–1)/÷(1-√2)

=(2–2√2)/(1-√2)=2(1-√2)/(1-√2)=2

other procedure,

√x-1/√x=(x-1)/√x=(3–2√2–1)/√x= (2–2√2)/√x

=2(1-√2)/√x=a ( say) , a^2=4(1-√2)^2/x

=4(1–2√2+2)/(3–2√2) = 4(3–2√2)/(3–2√2)=4

a=2 , √x-1/√x=2 ans

((√x)-(1/√x))^2=x+(1/x)-2*x*(1/x).I.e

(a-b)^2=a*a+b*b-2*a*b.

((√x)-(1/√x))^2=(3–2√2)+(1/3–2√2)-2*(3–2√2)*(1/3–2√2).

Rationalising the above factor (3–2√2) I.e

Multiplying with (3+2√2) in numerator and denominator.so the result be (3+2√2).

((√x)-(1/√x))^2=(3–2√2)+(3+2√2)-2.

((√x)-(1/√x))^2=3+3–2=4.

Applying square root on both sides

(√x)-(1/√x)=2.

So the answer is 2.

If x=17+12√2, what is the value of √x-1/√x?

If x=2+√3, then find the value of √(2x) +1/√(2x)?

If x= 3+2√2, what is the value of x^2+1/x^2?

If x=7+4√3, then what will be the value of √x+1/√x?

If x = 4/ (2√3+3√2), then what is the value of x+1/x?

x=3–22–√

x−−√=3–22–√−−−−−−√=2+1–22∗1−−−−√−−−−−−−−−−−√=2–√–1

1x=12–√–1=2–√+1

x−−√−1x=(2–√–1)−(2–√+1)=−2

x=3-2√2 so then it can be written as x=sqrt(2)^2 +sqrt(1)^2-2√2=(sqrt(2)+1)^2

so means sqrt(x) = sqrt (2) +1;

sqrt(x)-1/sqrt(x): (sqrt(2)+1 ) - (1/sqrt(2)+1)

rationalize it;

so the final answer is sqrt(2) +1 -sqrt(2) +1:

answer is 2.

{(√x) - (1/√x)}^2 = x + 1/x - 2

x= 3–2√2

1/x= 3+2√2 (on rationalising)

put the values in above equation

3–2√2+3+2√2–2

=4

{(√x) - (1/√x)}^2 = 4

taking root -2,+2

-2

X is greater than 0..Means positive

X can be written as= √2^2+1^2–2.√2.1=(√2–1)^2

√x=±(√2–1)…

But given x is positive so it's root should nt be negative ..

√x=√2–1

1/√x=1/(√2–1)

Rationalizing gives us √2+1

√x-√1/x=-2

Here it is…..

Hope it helps……

x= 1+ 2–√2−2∗1∗2–√=(1−2–√)2

thus x−−√ = 2–√−1

After rationalising, 1/ x−−√ = 2–√+1

thus, 1/ x−−√+x−−√=2∗2–√

The positive square root of3-2 √2 is √2 – 1 & reciprocal of √2 -1 is √2 + 1.

Hence √x – ( 1/√x) is --2

x = 3 - 2√2

=> √x = √2 - 1

=> 1/√x = 1/(√2 - 1)

=> 1/√x = √2 + 1

Now ,

√x - 1/√x = √2 - 1 - (√2 + 1)

=> √x - 1/√x = √2 - 1 - √2 - 1

=> √x - 1/√x = - 2

√x- (1\√x)

Mult by (√x÷√x)

1\√x (x-1)

2–2√2\ √x

2–2√2\ √(3–2√2)

If x=√7+4√3, then what us the value of [x+1\x]?

Given :

x=7–√+43–√

⟹1x=17–√+43–√

⟹1x=1(7–√+43–√)(7–√−43–√)(7–√−43–√)

(a+b)(a−b)=a2−b2

Therefore,

⟹1x=7–√−43–√(7–√)2−(43–√)2

⟹1x=7–√−43–√−41

⟹x+1x=7–√+43–√−7–√−43–√41

⟹x+1x=417–√+1643–√−7–√+43–√41

⟹x+1x=857–√+213–√41

Exact answer :

x+1x=857–√+213–√41

Decimal Approximation :

x+1x=9.67840488150122...

Thank You!

If x= (3-2√2), how do you show that (√x-1/√x) =2?

sqrt(x)-1/sqrt(x)=(x-1)/sqrt(x)=2

now square both sides and prove by substitution

hence [(x-1)^2}/x =4 or x^2–2x+1=4x

(3–2sqrt(2))^2 -2(3–2sqrt(2))+1=4(3–2qrt(2))

expanding

9–12sqrt(2)+8–6 +4sqrt(2) +1=12–8sqrt(2)

Collecting terms

12 -8sqrt(2)=12–8sqrt(2)

Hence 2 is a solution

Some interesting side notes:

The sqrt[3–2sqrt(2)]=[sqrt(2)-1] or [1-sqrt(2)]

If you use EXCEL and evaluate SQRT[(3–2*SQRT(2)], the answer = -2 ???

If you use the -SQRT() value for both terms the result will be 2. The SQRT() of a number is both a + and - number and EXCEL only returns the positive value

If you plot the possible graphs

If x= 3+2√2, what is the value of x^2+1/x^2?

x=3+22–√1x=13+22–√=3−22–√9−8=3−22–√(x2+1x2)=(x+1x)2−2(x2+1x2)=(3+22–√+3−23–√)2−2(x2+1x2)=36−2=34

If x=2+√3, then find the value of √(2x) +1/√(2x)?

Thanks for your attention.

Don't forget to upvote if it is helpful.

If X+(1/X)=7, what is the value of X√X+(1/X√x)?

Hi,

If you know formula of (a+b)^3 then you can easily solve this problem.

Ok i am solving

Let's start

Given

x+1/x=7

And we want to find x√x+1/x√x=?

ok

First if we will do cube of (x+1/x)^3 and then we will get

(x+1/x)^3= x^3+(1/x)^3+2*x*(1/x){x+1/x}=7^3

After that we get

x^3+(1/x)^3+2(x+1/x)=343

We know that value of x+1/x=7

Put value and get

x^3+1/x^3+2*7=343

x^3+1/x^3=343–14=329

Now

(x√x+1/x^x)^2= x^3+1/x^3+2

=329+2=331 ans.

If x=3+2√3, then what is the value of √x+1/√x?

Continued…

a^2 = (12+8√3)/3

a^2 = 4(3+2√3)/3

a = + or - 2 square root of [ (3+2√3)/3]

If x=3+2√2, then what is the value of √x-1/√x?

If ‘x=3-2√2’, can you find the value of ‘√x+1/√x’?

Answer: 2√2

Procedure: x = 3 -2√2 = (1)^2 + (√2)^2 - 2(1)(√2) =

(√2 - 1)^2

Therefore, √x = √2 - 1

Hence, √x + 1/√x = (√2 - 1) + 1/(√2 -1)

= (√2 - 1) + (√2 + 1)

= 2√2

Hope this Helps.

Answer 2

given

$x = 3 + 2 \sqrt{2}$

so we can write

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }$

rationalizing denominator we get

$\frac{1}{x} = \frac{3 - 2 \sqrt{2} }{9 - 8} \\ \\ \frac{1}{x} = 3 - 2 \sqrt{2}$

Now

$x + \frac{1}{x} = 3 + 2 \sqrt{2} + 3 - 2 \sqrt{2} \\ \\ x + \frac{1}{x} = 6$