Solve the equations:
`10x^(-2)-9-x^(-4)=0`.

Answer

To solve the equation \( 10x^{-2} - 9 - x^{-4} = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the terms with negative exponents as fractions:
\[ 10x^{-2} = \frac{10}{x^2}, \quad x^{-4} = \frac{1}{x^4} \]
Thus, the equation becomes:
\[ \frac{10}{x^2} - 9 - \frac{1}{x^4} = 0 \]

### Step 2: Multiply through by \( x^4 \)
To eliminate the fractions, we multiply the entire equation by \( x^4 \) (assuming \( x \neq 0 \)):
\[ 10x^2 - 9x^4 - 1 = 0 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ -9x^4 + 10x^2 - 1 = 0 \]
or equivalently,
\[ 9x^4 - 10x^2 + 1 = 0 \]

### Step 4: Substitute \( y = x^2 \)
Let \( y = x^2 \). Then the equation becomes a quadratic in \( y \):
\[ 9y^2 - 10y + 1 = 0 \]

### Step 5: Use the quadratic formula
We can solve for \( y \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 9 \), \( b = -10 \), and \( c = 1 \):
\[ y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 1}}{2 \cdot 9} \]
\[ y = \frac{10 \pm \sqrt{100 - 36}}{18} \]
\[ y = \frac{10 \pm \sqrt{64}}{18} \]
\[ y = \frac{10 \pm 8}{18} \]

### Step 6: Calculate the values of \( y \)
Calculating the two possible values for \( y \):
1. \( y = \frac{10 + 8}{18} = \frac{18}{18} = 1 \)
2. \( y = \frac{10 - 8}{18} = \frac{2}{18} = \frac{1}{9} \)

### Step 7: Back-substitute for \( x \)
Since \( y = x^2 \), we have:
1. \( x^2 = 1 \) gives \( x = \pm 1 \)
2. \( x^2 = \frac{1}{9} \) gives \( x = \pm \frac{1}{3} \)

### Final Solution
Thus, the solutions for \( x \) are:
\[ x = 1, \quad x = -1, \quad x = \frac{1}{3}, \quad x = -\frac{1}{3} \]