Problem 44 - 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 expression simplifies to \(-\frac{1}{4x^2}\) with positive exponents.

Step 1: Understanding Negative Exponents

A negative exponent indicates reciprocal. For an expression like \((a)^{-b}\), it means \(\frac{1}{a^b}\). Thus, \((2x)^{-2}\) becomes \(\frac{1}{(2x)^2}\).

Step 2: Simplifying the Expression

Since \((2x)^2 = 2^2 \cdot x^2 = 4x^2\), then \(\frac{1}{(2x)^2} = \frac{1}{4x^2}\).

Step 3: Applying the Negative Sign

The original expression is \(-(2x)^{-2}\). We have \(\frac{1}{4x^2}\) from the previous step, and applying the negative sign, we get \(-\frac{1}{4x^2}\). This is the expression with positive exponents.

Key Concepts

Reciprocal of Exponents

When working with exponents, encountering a negative exponent might seem tricky at first, but it's simply a signal to take the reciprocal of the base.
For instance, if we have \((a)^{-b}\), this translates to \(\frac{1}{a^b}\). Therefore, the sign of the exponent tells us to "flip" the base into a fraction. To put it simply:

• \((a)^{-b}\) = \(\frac{1}{a^b}\)
• This conversion into a reciprocal allows us to work with positive exponents, which are usually easier to handle when simplifying expressions.

Using this method ensures that all parts of the expression are manageable and consistent, especially when dealing with variables and numerical coefficients together.

Positive Exponents

A positive exponent indicates how many times the base is multiplied by itself. For example, \((2)^3\) means that the number 2 is multiplied by itself three times, resulting in 8.
Reactive positive exponents can simplify complex calculations and enable clearer formulations of expressions.In the exercise at hand, the negative exponent becomes positive once the reciprocal is taken:

• The expression \((2x)^{-2}\) converts to \(\frac{1}{(2x)^2}\).
• This simplifies to \(\frac{1}{4x^2}\) as positive exponents make it easier to see how the number and variables interact.

Utilizing positive exponents in expressions generally leads to more straightforward algebraic manipulation and simplification, making calculations more intuitive.

Simplifying Expressions

One of the final steps in working with exponents is simplifying the expression, particularly when it’s asked to be presented in its simplest form.
Once we've handled the negative exponents by converting them into positive ones using reciprocals, the next goal is to simplify.Simplifying involves:

• Breaking down each part of the expression into its simplest form, like calculating \((2x)^2\) to get \(4x^2\).
• Ensuring any constants, coefficients, and variables are fully consolidated, reducing any fractions to their simplest terms where possible.

For example, in the original expression \(-(2x)^{-2}\), the simplification process transforms it to \(-\frac{1}{4x^2}\).
This shows the expression in its simplest positive exponent form, reflecting a cleaner and more concise mathematical reasoning.