91Ó°ÊÓ

In algebra, working with exponents is a key part of simplifying expressions. Exponents represent repeated multiplication, such as in the term \(n^{-2}\). Negative exponents like \(n^{-2}\) can be rewritten using positive exponents.
To transform \(n^{-2}\) into a positive exponent, you take the reciprocal, which means flipping it around to become \(\frac{1}{n^2}\).
This is a simple rule: **any negative exponent on a number can be turned positive by moving it to the denominator** (for instance, \(a^{-b} = \frac{1}{a^b}\)).
When you perform these transformations, it becomes much easier to handle other algebraic operations, as we've done with the expression \((n^{-2} - 2n^{-1})^2\) by turning it into \(\left(\frac{1}{n^2} - 2\cdot\frac{1}{n}\right)^2\). This transformation is a critical first step in simplifying expressions with exponents.

Fraction simplification is an important process, especially when dealing with algebra. Once we transform expressions such as \(n^{-1}\) into \(\frac{1}{n}\), the next step is to simplify these fractions.
In the example given, we started with the expression \(\left(\frac{1}{n^2} - 2\cdot\frac{1}{n}\right)^2\).
The process involves breaking down each fraction to its simplest form, combining them carefully. If fractions share a common denominator, our task becomes easier because we can work with the numerators directly. In our exercise:

• The fractions \(\frac{1}{n^2}\) and \(2\cdot\frac{1}{n}\) needed to be combined.
• By rewriting \(2\cdot\frac{1}{n}\) as \(\frac{2n}{n^2}\), the fractions can be easily combined into \(\frac{1 - 2n}{n^2}\).

Simplifying fractions in this manner helps reduce the complexity of algebraic expressions and makes them much more manageable in further steps.

Polynomial expansion is a crucial skill when dealing with expressions raised to a power. In our example, we need to expand \((1 - 2n)^2\).
The expansion process involves using the binomial theorem, which states:\[(a + b)^2 = a^2 + 2ab + b^2\]
Applying this to our expression:\

• Let \(a = 1\) and \(b = -2n\).
• Then \((1 - 2n)^2 = 1^2 - 2\cdot1\cdot2n + (-2n)^2\).
• This becomes \(1 - 4n + 4n^2\).

Through the binomial theorem, any polynomial raised to a power can be expanded methodically. This technique is not only helpful but necessary for simplifying complex expressions, as it allows for a clear breakdown of terms.

When adding or subtracting fractions, having a common denominator is essential. A common denominator is the shared multiple of the denominators of two or more fractions. It allows you to combine fractions seamlessly.
In our exercise, we needed to combine \(\frac{1}{n^2}\) and \(2\cdot\frac{1}{n}\), which originally had different denominators of \(n^2\) and \(n\), respectively.

• The least common denominator here is \(n^2\).
• This means we could rewrite \(2\cdot\frac{1}{n}\) as \(\frac{2n}{n^2}\) to match the denominator of \(\frac{1}{n^2}\).
• This transformation results in \(\frac{1 - 2n}{n^2}\).

By ensuring expressions share a common denominator, it simplifies the process and makes it convenient to perform further operations like polynomial expansions and simplifications.