Problem 82 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. $$ \frac{1}{(2 x)^{-2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(4x^2\)

Step 1: Identify the expression

The expression to be simplified is \( \frac{1}{(2x)^{-2}} \)

Step 2: Use the exponent rule

The rule \(a^{-n} = \frac{1}{a^n}\) can be used to simplify this expression. Apply this to the given expression, \( (2x)^{-2} \) becomes \( \frac{1}{(2x)^2} \). So, the expression becomes \( \frac{1}{\frac{1}{(2x)^2}} \)

Step 3: Simplify the fraction

This is a fraction of fractions. Simplify by multiplying the numerator and the denominator of the outer fraction by \((2x)^2\). This will remove the denominator. So, the expression simplifies to \((2x)^2\)

Step 4: Expand the expression

The expression \((2x)^2\) can be expanded to \(4x^2\) so this is the simplest form of the expression

Key Concepts

Exponent Rules

Understanding exponent rules is vital in simplifying expressions like \( \frac{1}{(2x)^{-2}} \). Let's delve into the basics:

• Negative Exponents: The rule \( a^{-n} = \frac{1}{a^n} \) tells us that a negative exponent means "one over" the base raised to the positive of that exponent.
• Power of a Product: This rule states that \( (ab)^n = a^n b^n \). It means you can "distribute" the exponent to both factors inside the parentheses.

In our example \((2x)^{-2}\), the negative exponent indicates that we need to take the reciprocal and change the power to positive, turning it into \(\frac{1}{(2x)^2}\). By applying the power of a product rule, you can further break down \((2x)^2\) to \(2^2 \cdot x^2\). This simplifies to \(4x^2\) after calculations.Knowing these rules helps you deal with exponents confidently without confusion.

Fraction Simplification

Fraction simplification involves turning complex fractions into simpler ones. In the problem \( \frac{1}{(2x)^{-2}} \), we encounter a fraction where the denominator is also a fraction, \(\frac{1}{(2x)^2}\). This is often called a fraction of fractions.

• Remove Fraction in Denominator: To simplify, multiply both the numerator and the denominator of the outer fraction by \((2x)^2\). This eliminates the fraction in the denominator and simplifies the whole expression.
• Resulting Simplification: When you multiply \(1\) by \((2x)^2\), it results in \((2x)^2\), effectively removing any fractions.

Removing fractions is all about multiplying by what's necessary to cancel out the complex parts, leaving you with a neat and clean expression. This step makes the expression more comfortable to work with and understand.

Algebraic Expressions

Algebraic expressions like \( (2x)^2 \) involve variables and numbers combined through operations. Let's explore how to deal with them:

• Expand the Expression: To expand \((2x)^2\), use the distributive property \((a+b)^n = a^n + b^n\). Apply this to both parts: \((2)^2 \times (x)^2\) results in \(4x^2\).
• Combining Like Terms: Though less relevant in this specific problem, keep in mind that combining like terms simplifies expressions further. This technique groups similar terms, like \(x^2\) terms to consolidate computations.

Understanding algebraic expressions involves breaking them into manageable parts using fundamental properties like distribution. The process of simplifying ensures the result is as straightforward as possible, making it easier to handle further algebraic tasks.