Question

Simplify, and write without negative exponents. Do some by calculator. $$4 a^{-2}$$

Short answer

\(\frac{4}{a^2}\)

Step-by-step solution

Identify Negative Exponents

The expression has a negative exponent, which indicates that the variable (in this case 'a') needs to be moved to the denominator to make the exponent positive.

Rewrite with Positive Exponents

To convert a negative exponent into a positive one, you take the reciprocal of the base. For example, \(a^{-2}\) becomes \(\frac{1}{a^2}\).

Combine Terms

Multiply the numerical coefficient by the new expression with a positive exponent to get the final simplified expression. Here, \(4 \times \frac{1}{a^2} = \frac{4}{a^2}\).

Key concepts

Simplifying Expressions

When it comes to algebra, one of the most useful skills you can have is simplifying expressions. To simplify means to make something less complicated or easier to understand. In mathematical terms, simplifying an expression is all about reducing it to its most basic form, without changing its value.

For instance, when faced with an expression like \(4a^{-2}\), simplifying it involves getting rid of any negative exponents because they can make expressions appear more complex than they really are. The simplified form should be as straightforward as possible, so it's clear what the expression really represents.

To accomplish this, you'll need to understand and apply specific laws of exponents effectively. By mastering these rules, you can simplify expressions confidently, making them easier to work with whether you're solving equations, graphing functions, or working on other mathematical problems. It's a crucial step in any algebraic process, as a simplified expression often reveals the meaning and relationships inherent in the mathematical statement.

Reciprocal of Exponents

Negative exponents can sometimes be a source of confusion, but their treatment is fairly standard in algebra: they indicate the reciprocal of the base raised to the absolute value of the expressed exponent. The reciprocal of a number involves switching the numerator and denominator if you think of the number as a fraction. For example, the reciprocal of \(2\) is \(\frac{1}{2}\), and the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

The same applies to variables with exponents. When encountering a negative exponent, you are essentially looking at the reciprocal of the base with a positive exponent. So, for the given exercise where we have \(4a^{-2}\), the negative exponent tells us to take the reciprocal of \(a^2\), making it \(\frac{1}{a^2}\). The 4 remains as the numerator, so after applying the concept of reciprocal of exponents, our expression simplifies to \(\frac{4}{a^2}\).

This approach allows us to translate any expression with negative exponents into a more conventional and familiar form, one that aligns with our intuitive understanding of division and fractions.

Algebraic Manipulation

Algebraic manipulation is a broad term that covers various techniques used to rearrange, simplify, and solve algebraic expressions and equations. It's the art of using rules and operations to transform complex problems into simpler ones that are easier to interpret or solve. This includes operations such as adding, subtracting, multiplying, and dividing variables and constants, along with applying the rules of exponents.

When you're given an expression like \(4a^{-2}\), using algebraic manipulation, you're expected not only to apply the rules of exponents but also to perform the correct order of operations to ensure the expression is simplified correctly. The algebraic manipulation process often involves several steps, such as those outlined in our exercise's solution:

• Identifying the negative exponents.
• Rewriting them as reciprocals with positive exponents.
• Combining like terms and coefficients wisely to reach the final simplified form.

Through practice and a clear understanding of algebraic rules, you'll become more adept at manipulating expressions like these swiftly and accurately, an essential skill in any field that requires a solid foundation in algebra.