Solve If x ^ 8 + 1/(x ^ 8) = a then find find x + 1/x ii ) x ^ 2 - 1/(x ^ 2) iii ) x ^ 4 - 1/(x ^ 4) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x ^ { 8 } + \frac { 1 } { x ^ { 8 } } = a A \times + \frac { 1 } { x } \quad x ^ { 2 } - \frac { 1 } { x ^ { 2 } } \quad x ^ { 4 } - \frac { 1 } { x ^ { 4 } }$$

Solve for A (complex solution)

$\left\{\begin{matrix}A=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}\text{, }&x\neq 0\text{ and }a\neq 0\\A\in \mathrm{C}\text{, }&-x^{16}-x^{10}-x^{4}-1=0\text{ and }a=0\text{ and }x\neq 0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $x^{8}$, the least common multiple of $x^{8},x,x^{2},x^{4}$.

$$x^{8}x^{8}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

To multiply powers of the same base, add their exponents. Add $8$ and $8$ to get $16$.

$$x^{16}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $a\times \frac{1}{x}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{a}{x}Ax^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{a}{x}x^{2}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{ax^{2}}{x}A-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Cancel out $x$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{1}{x^{2}}x^{4}$ as a single fraction.

$$x^{16}+1=x^{8}\left(axA-\frac{x^{4}}{x^{2}}\right)-x^{4}$$

Cancel out $x^{2}$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-x^{2}\right)-x^{4}$$

Use the distributive property to multiply $x^{8}$ by $axA-x^{2}$.

$$x^{16}+1=aAx^{9}-x^{10}-x^{4}$$

Swap sides so that all variable terms are on the left hand side.

$$aAx^{9}-x^{10}-x^{4}=x^{16}+1$$

Add $x^{10}$ to both sides.

$$aAx^{9}-x^{4}=x^{16}+1+x^{10}$$

Add $x^{4}$ to both sides.

$$aAx^{9}=x^{16}+1+x^{10}+x^{4}$$

The equation is in standard form.

$$ax^{9}A=x^{16}+x^{10}+x^{4}+1$$

Divide both sides by $ax^{9}$.

$$\frac{ax^{9}A}{ax^{9}}=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}$$

Dividing by $ax^{9}$ undoes the multiplication by $ax^{9}$.

$$A=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}$$

Solve for a (complex solution)

$\left\{\begin{matrix}a=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}\text{, }&x\neq 0\text{ and }A\neq 0\\a\in \mathrm{C}\text{, }&-x^{16}-x^{10}-x^{4}-1=0\text{ and }A=0\text{ and }x\neq 0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $x^{8}$, the least common multiple of $x^{8},x,x^{2},x^{4}$.

$$x^{8}x^{8}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

To multiply powers of the same base, add their exponents. Add $8$ and $8$ to get $16$.

$$x^{16}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $a\times \frac{1}{x}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{a}{x}Ax^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{a}{x}x^{2}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{ax^{2}}{x}A-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Cancel out $x$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{1}{x^{2}}x^{4}$ as a single fraction.

$$x^{16}+1=x^{8}\left(axA-\frac{x^{4}}{x^{2}}\right)-x^{4}$$

Cancel out $x^{2}$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-x^{2}\right)-x^{4}$$

Use the distributive property to multiply $x^{8}$ by $axA-x^{2}$.

$$x^{16}+1=aAx^{9}-x^{10}-x^{4}$$

Swap sides so that all variable terms are on the left hand side.

$$aAx^{9}-x^{10}-x^{4}=x^{16}+1$$

Add $x^{10}$ to both sides.

$$aAx^{9}-x^{4}=x^{16}+1+x^{10}$$

Add $x^{4}$ to both sides.

$$aAx^{9}=x^{16}+1+x^{10}+x^{4}$$

The equation is in standard form.

$$Ax^{9}a=x^{16}+x^{10}+x^{4}+1$$

Divide both sides by $Ax^{9}$.

$$\frac{Ax^{9}a}{Ax^{9}}=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}$$

Dividing by $Ax^{9}$ undoes the multiplication by $Ax^{9}$.

$$a=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}$$

Solve for A

$A=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}$
$x\neq 0\text{ and }a\neq 0$

Steps for Solving Linear Equation

Multiply both sides of the equation by $x^{8}$, the least common multiple of $x^{8},x,x^{2},x^{4}$.

$$x^{8}x^{8}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

To multiply powers of the same base, add their exponents. Add $8$ and $8$ to get $16$.

$$x^{16}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $a\times \frac{1}{x}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{a}{x}Ax^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{a}{x}x^{2}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{ax^{2}}{x}A-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Cancel out $x$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{1}{x^{2}}x^{4}$ as a single fraction.

$$x^{16}+1=x^{8}\left(axA-\frac{x^{4}}{x^{2}}\right)-x^{4}$$

Cancel out $x^{2}$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-x^{2}\right)-x^{4}$$

Use the distributive property to multiply $x^{8}$ by $axA-x^{2}$.

$$x^{16}+1=aAx^{9}-x^{10}-x^{4}$$

Swap sides so that all variable terms are on the left hand side.

$$aAx^{9}-x^{10}-x^{4}=x^{16}+1$$

Add $x^{10}$ to both sides.

$$aAx^{9}-x^{4}=x^{16}+1+x^{10}$$

Add $x^{4}$ to both sides.

$$aAx^{9}=x^{16}+1+x^{10}+x^{4}$$

The equation is in standard form.

$$ax^{9}A=x^{16}+x^{10}+x^{4}+1$$

Divide both sides by $ax^{9}$.

$$\frac{ax^{9}A}{ax^{9}}=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}$$

Dividing by $ax^{9}$ undoes the multiplication by $ax^{9}$.

$$A=\frac{x^{16}+x^{10}+x^{4}+1}{ax^{9}}$$

Solve for a

$a=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}$
$x\neq 0\text{ and }A\neq 0$

Steps for Solving Linear Equation

Multiply both sides of the equation by $x^{8}$, the least common multiple of $x^{8},x,x^{2},x^{4}$.

$$x^{8}x^{8}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

To multiply powers of the same base, add their exponents. Add $8$ and $8$ to get $16$.

$$x^{16}+1=x^{8}\left(aA\times \frac{1}{x}x^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $a\times \frac{1}{x}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{a}{x}Ax^{2}-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{a}{x}x^{2}$ as a single fraction.

$$x^{16}+1=x^{8}\left(\frac{ax^{2}}{x}A-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Cancel out $x$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-\frac{1}{x^{2}}x^{4}\right)-x^{4}$$

Express $\frac{1}{x^{2}}x^{4}$ as a single fraction.

$$x^{16}+1=x^{8}\left(axA-\frac{x^{4}}{x^{2}}\right)-x^{4}$$

Cancel out $x^{2}$ in both numerator and denominator.

$$x^{16}+1=x^{8}\left(axA-x^{2}\right)-x^{4}$$

Use the distributive property to multiply $x^{8}$ by $axA-x^{2}$.

$$x^{16}+1=aAx^{9}-x^{10}-x^{4}$$

Swap sides so that all variable terms are on the left hand side.

$$aAx^{9}-x^{10}-x^{4}=x^{16}+1$$

Add $x^{10}$ to both sides.

$$aAx^{9}-x^{4}=x^{16}+1+x^{10}$$

Add $x^{4}$ to both sides.

$$aAx^{9}=x^{16}+1+x^{10}+x^{4}$$

The equation is in standard form.

$$Ax^{9}a=x^{16}+x^{10}+x^{4}+1$$

Divide both sides by $Ax^{9}$.

$$\frac{Ax^{9}a}{Ax^{9}}=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}$$

Dividing by $Ax^{9}$ undoes the multiplication by $Ax^{9}$.

$$a=\frac{x^{16}+x^{10}+x^{4}+1}{Ax^{9}}$$