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

Square Both Sides: We are given that x + (1)/(x) = (2)/(3). To find x^(2) + (1)/(x^(2)), we first need to square both sides of the given equation to find an expression for x^(2) + (1)/(x^(2)). (x + (1)/(x))^2 = ((2)/(3))^2Expand 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*(x)*(1/x) + (1/x)^2 = (2/3)^2Simplify Equation: Simplify the middle term and the right side of the equation: x^2 + 2 + (1/x)^2 = 4/9Isolate 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: x^2 + (1/x)^2 = (4/9) - 2Combine Fractions: Convert 2 to a fraction with a denominator of 9 to combine it with 4/9: x^2 + (1/x)^2 = (4/9) - (18/9)Subtract Fractions: Subtract the fractions: x^2 + (1/x)^2 = -14/9Final Result: Now we have the value of x^2 + (1/x)^2, which is -14/9.

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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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Algebra 1

Evaluate integers raised to rational exponents

Full solution

\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 $-14/9$.