Simplify.
$\frac{b+a^{-2}}{a-b^{-2}}$
Write your answer using only positive exponents.
The expression we are given is \(\frac{b+a^{-2}}{a-b^{-2}}\). The goal is to simplify this expression using only positive exponents. We assume that \(a
eq 0\) and \(b
eq 0\) so that the negative exponents and the division are well-defined. Negative exponents are defined by the rule \(x^{-n} = \frac{1}{x^n}\). Therefore, we can rewrite \(a^{-2}\) as \(\frac{1}{a^2}\) and \(b^{-2}\) as \(\frac{1}{b^2}\). The expression becomes:
\[ \frac{b + \frac{1}{a^2}}{a - \frac{1}{b^2}} \]
To simplify this, we rewrite the numerator and the denominator as single fractions.
For the numerator:
\[ b + \frac{1}{a^2} = \frac{b \cdot a^2}{a^2} + \frac{1}{a^2} = \frac{a^2b + 1}{a^2} \]
For the denominator:
\[ a - \frac{1}{b^2} = \frac{a \cdot b^2}{b^2} - \frac{1}{b^2} = \frac{ab^2 - 1}{b^2} \]
Thus, the entire expression can be written as:
\[ \frac{\frac{a^2b + 1}{a^2}}{\frac{ab^2 - 1}{b^2}} \]
To divide by a fraction, we multiply by its reciprocal. Therefore:
\[ \frac{a^2b + 1}{a^2} \cdot \frac{b^2}{ab^2 - 1} = \frac{b^2(a^2b + 1)}{a^2(ab^2 - 1)} \]
Distributing the terms, we get:
\[ \frac{a^2b^3 + b^2}{a^3b^2 - a^2} \]
This is the simplified form using only positive exponents.
\[ \boxed{\frac{b^2(a^2b + 1)}{a^2(ab^2 - 1)}} \]
Simplifying algebraic expressions with negative exponents is essential in fields like physics and engineering, where complex formulas often represent physical properties. For example, when calculating resistance in a circuit or gravitational force, you might encounter expressions with varied exponents that need to be simplified to a standard, clean form to perform further analysis or to program a controller efficiently. Learning to handle these algebraic manipulations allows you to manage such data more effectively, ensuring that your calculations remain organized and accurate.