Problem 41 - 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

\(\frac{1}{4x^2}\)

Step 1: Understanding Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In general, for any nonzero number \(a\), \(a^{-n} = \frac{1}{a^n}\).

Step 2: Applying the Negative Exponent Rule

Apply the rule from Step 1 to the expression \((2x)^{-2}\). This means we write it as the reciprocal with a positive exponent: \((2x)^{-2} = \frac{1}{(2x)^2}\).

Step 3: Expanding the Squared Expression

Expand \((2x)^2\) by applying the exponent to both the number and the variable inside the parentheses: \((2x)^2 = 2^2 \cdot x^2 = 4x^2\).

Step 4: Writing the Final Expression

Substitute the expanded expression from Step 3 back into the reciprocal form: \(\frac{1}{4x^2}\). This is the expression with only positive exponents.

Key Concepts

Reciprocal

A reciprocal is a concept that involves flipping a fraction or turning a whole number into one. When you have a number and you want to find its reciprocal, you essentially take one and divide it by that number.
For instance, the reciprocal of 5 is \( \frac{1}{5} \). Similarly, for a fraction like \( \frac{3}{4} \), the reciprocal is simply reversed, becoming \( \frac{4}{3} \).

• A reciprocal changes when dealing with negative exponents. A negative exponent, like \( a^{-n} \), tells us to take the reciprocal of the base: \( a^{-n} = \frac{1}{a^n} \).
• This means if you see a negative exponent, you know your first step is to rewrite it as a reciprocal and alter the negative exponent to a positive.

Applying this principle to oindent \( (2x)^{-2} \), we change it to the reciprocal: \( \frac{1}{(2x)^2} \). This lays the foundation for further simplification.

Positive Exponents

Exponents are a way to show repeated multiplication of a number by itself. When we talk about positive exponents, we are referring to the natural way of expressing this repetition.
A positive exponent \( n \) means: multiply the base \( n \) times. For example, \( 2^3 \) means \( 2 \times 2 \times 2 = 8 \).

• When changing from negative to positive exponents, our focus is moving from the idea of reciprocals (expressions of division) back to multiplication.
• In our current discussion, converting \( (2x)^{-2} \) involves switching to \( (2x)^2 \), changing the nature of operations from division to repeated multiplication.

After applying this conversion, we find \( (2x)^2 = 2^2 \times x^2 \), giving us \( 4x^2 \). Using only positive exponents makes expressions easier to work with and more intuitive for further calculations.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that makes mathematical expressions easier to interpret and work with. The goal is to reduce the expression to its most basic form without changing its value.
Here's how you can tackle simplification:

• Start by dealing with the exponents, as seen in relocating negative exponents to positive ones by using their reciprocals.
• Multiply out factors when necessary. In our example, \( (2x)^2 \) was expanded to \( 4x^2 \) because \( 2^2 = 4 \) and \( x^2 \) is simply \( x \times x \).
• Combine any similar terms and write the expression in a manner that clearly indicates its simplest form.

The step of simplifying \( (2x)^{-2} \) led us to the final form \( \frac{1}{4x^2} \). It's crucial to systematically follow rules such as these to achieve the simplest form of any algebraic expression.