Problem 46 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ (-2 x)^{-2} $$

Step-by-Step Solution

Answer

The answer is \(\frac{1}{4x^2}.\)

Step 1: Apply the Rule for Negative Exponents

Any term with a negative exponent can be rewritten as its reciprocal with a positive exponent. For the expression \((-2x)^{-2}\), we can rewrite this as follows: \((-2x)^{-2} = \frac{1}{(-2x)^2}.\)

Step 2: Simplify the Expression Inside the Parentheses

Now, compute \((-2x)^2\). Remember to square each part of the expression:\((-2x)^2 = (-2)^2 \times x^2 = 4x^2.\) Thus, the expression becomes \(\frac{1}{4x^2}.\)

Step 3: Write the Final Answer in Simplest Form

The expression with only positive exponents, in its simplest form, is \(\frac{1}{4x^2}.\)

Key Concepts

Reciprocal

In mathematics, understanding the concept of a reciprocal is an essential skill, particularly when dealing with negative exponents. The reciprocal of a number or expression is simply one divided by that number or expression. For instance, if we have a number \(a\), its reciprocal is \(\frac{1}{a}\). This concept is crucial when working with negative exponents as it allows us to transform them into positive exponents.

Suppose we come across a term such as \((-2x)^{-2}\). The negative exponent indicates that we want the reciprocal of \((-2x)^2\), which can be rewritten as \(\frac{1}{(-2x)^2}\). This transformation is vital because it enables us to work with more manageable positive exponents. By flipping the expression to its reciprocal form, we ensure all exponents in our expression are positive, making further mathematical operations more straightforward.

Simplifying Expressions

Simplifying expressions is a fundamental process in algebra that involves rewriting expressions in their most concise and understandable form. This process often requires multiple steps, such as distributing powers and combining like terms.

In the context of our expression \(\frac{1}{(-2x)^2}\), simplifying involves squaring each element inside the parentheses. This means we'll find \((-2)^2\) and \(x^2\). Calculating these gives us:

• \((-2)^2 = 4\)
• \(x^2 = x^2\)

Thus, squared, the entire expression \((-2x)^2\) becomes \(4x^2\). Our simplified expression now reads \(\frac{1}{4x^2}\). This form is more accessible for further operations and gives us a clear path to understanding the entire problem.

Positive Exponents

Positive exponents are often more intuitive to work with than negative ones. They indicate the number of times a base is multiplied by itself. For example, the expression \(x^2\) means \(x\) times \(x\).

Transforming expressions to have only positive exponents is typically a goal in algebra because calculations can be done more smoothly and are less error-prone. When you deal with a negative exponent, like in our original terms \((-2x)^{-2}\), rewriting it as a positive exponent involves finding the reciprocal, resulting in \(\frac{1}{(-2x)^2}\).

After simplifying what this means — as described earlier — the expression with positive exponents, \(\frac{1}{4x^2}\), is much easier to engage with. Positive exponents allow us to employ standard arithmetic operations without the added complexity negative exponents introduce.