# Expand: 3(1-3x)^-1 + 4(2+x)^-2

• thomas49th
In summary, expanding an equation means to simplify or rewrite it in a more expanded form, often by expanding the terms within parentheses. To expand an equation with negative exponents, we use the rule a^(-n) = 1/a^n. The purpose of expanding an equation is to make it easier to solve or manipulate. The steps to expand an equation include rewriting it, expanding each term using exponent rules, and distributing coefficients. While not necessary, expanding an equation can make it easier to work with and provide more insights into the underlying patterns or relationships between terms.

## Homework Statement

Expand in ascending powers of x upto and including the term x³:

$$3(1-3x)^{-1} + 4(2+x)^{-2}$$

## The Attempt at a Solution

Err never done it with 2 terms like above before. Any suggestions/hints

Thanks :)

expand $(1-3x)^{-1}$ up to the term in $x^3$ and expand $(2+x)^{-2}$ as well up to the term in $x^3$, the just add them.

## 1. What does "Expand" mean in this equation?

"Expand" means to simplify or rewrite the given equation in a more expanded form. In this equation, we are asked to expand the given expression by expanding the terms within the parentheses.

## 2. How do you expand an equation with negative exponents?

To expand an equation with negative exponents, we use the rule that states a^(-n) = 1/a^n. In this equation, we can rewrite the expressions within the parentheses as fractions with positive exponents before expanding.

## 3. What is the purpose of expanding an equation?

The purpose of expanding an equation is to make it easier to solve or manipulate. By expanding an equation, we can see all the individual terms and simplify them in a step-by-step manner.

## 4. Can you explain the steps to expand this equation?

First, we rewrite the given equation as 3/(1-3x) + 4/(2+x)^2. Then, we expand the first term using the rule mentioned in question 2 and get 3/(1-3x) = 3(1-3x)^(-1). Similarly, we expand the second term as 4/(2+x)^2 = 4(2+x)^(-2). Finally, we can distribute the coefficients to the terms within the parentheses to get the expanded form.

## 5. Is it necessary to expand this equation?

No, it is not necessary to expand this equation. However, expanding it can make it easier to work with, especially if we need to simplify or solve the equation further. In some cases, expanding an equation may also provide more insights into the underlying patterns or relationships between the terms.