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Find the value of
x^(2)+(1)/(x^(2)) when
x+(1)/(x)=(2)/(3),

$\newline$ Find the value of $x^{2}+\frac{1}{x^{2}}$ when $x+\frac{1}{x}=\frac{2}{3}$ ,

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Evaluate integers raised to rational exponents

Full solution

Q.

\newline

Find the value of

x^{2}+\frac{1}{x^{2}}

when

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

,

Square Both Sides: We are given that $x + \frac{1}{x} = \frac{2}{3}$ . To find $x^{2} + \frac{1}{x^{2}}$ , we first need to square both sides of the given equation to find an expression for $x^{2} + \frac{1}{x^{2}}$ . $\newline$ $(x + \frac{\(1\)}{x})^\(2\) = (\frac{\(2\)}{\(3\)})^\(2\)
Expand Left Side: Expanding the left side of the equation using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we get:\(x^2 + 2\cdot(x)\cdot(1/x) + (1/x)^2 = (2/3)^2\)
Simplify Equation: Simplify the middle term and the right side of the equation: \(x^2 + 2 + \left(\frac{1}{x}\right)^2 = \frac{4}{9}\)
Isolate \(x^2 + (1/x)^2\): We are looking for \(x^2 + (1/x)^2\), which is almost the left side of the equation we have, except for the extra \(2\). To isolate \(x^2 + (1/x)^2\), we subtract \(2\) from both sides of the equation:\(\newline\)\[x^2 + (1/x)^2 = (4/9) - 2\]
Combine Fractions: Convert \(2\) to a fraction with a denominator of \(9\) to combine it with \(\frac{4}{9}\):\(\newline\)\[x^2 + \left(\frac{1}{x}\right)^2 = \left(\frac{4}{9}\right) - \left(\frac{18}{9}\right)\]
Subtract Fractions: Subtract the fractions: \(x^2 + \left(\frac{1}{x}\right)^2 = -\frac{14}{9}\)
Final Result: Now we have the value of \(x^2 + (1/x)^2\), which is \(-14/9\).

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