Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\left(4 x^{2}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified expression of \((4 x^{2})^{-2}\) is \(\frac{1}{{16x^{4}}}\).

Step 1: Understand the Components

In the given expression \((4 x^{2})^{-2}\), there's a negative exponent -2 applied to the bracket \(4 x^{2}\). According to exponent rules, any number or term with a negative exponent, means taking the reciprocal of the base to the power of the absolute value of the exponent.

Step 2: Apply Negative Exponents Rule

The negative exponent has to be made positive. Therefore, \((4 x^{2})^{-2}\) will be re-written as \(\frac{1}{{(4x^{2})^{2}}}\). The negative exponent has been eliminated.

Step 3: Apply Exponents of Exponents Rule

Now we apply the exponent of exponent rule which states that \((a^{m})^{n} = a^{m*n}\). Using this, we simplify the expression further to \(\frac{1}{{16x^{4}}}\).

Key Concepts

Negative Exponents

Negative exponents might seem confusing at first, but they follow a simple rule that makes them easy to manage. A negative exponent indicates that you should take the reciprocal of the base and then raise it to the positive of that exponent. For example, if you have a term like \(a^{-b}\), it can be rewritten as \(\frac{1}{a^b}\). This means you flip the term upside down, and the exponent becomes positive.

• This concept helps in simplifying expressions because it allows you to eliminate negative signs in the exponents by using reciprocals.
• In more practical terms, it transforms a multiplication problem into a division problem, which can sometimes be easier to handle.

Understanding how to maneuver around negative exponents can greatly aid in algebraic simplification tasks.

Exponent of Exponent Rule

The exponent of exponent rule comes into play when you have an expression where an exponent is raised to another exponent, such as \((a^m)^n\). This rule simplifies expressions by allowing you to multiply the exponents: \(a^{m \times n}\).

• This is essential when you encounter nested exponentials, as it allows direct and efficient simplification.
• Applying this rule helps prevent errors that often occur from computing such expressions without simplifying first.

In our example, once the negative exponent was dealt with, the expression became easier to simplify. This rule changes complex expressions into simple, single exponentials quickly and accurately.

Reciprocals

Reciprocals are a fundamental concept in mathematics, often used in conjunction with negative exponents. The reciprocal of a number is essentially "flipping" it. For a number \(a\), its reciprocal is \(\frac{1}{a}\).

• Reciprocals are especially handy when dividing fractions or simplifying algebraic expressions with negative exponents.
• In equations, they help simplify calculations by turning division problems into multiplication, given that \(a \times \frac{1}{a} = 1\).

Utilizing reciprocals becomes a crucial step in simplifying expressions like \((4x^{2})^{-2}\) by turning it into a more manageable form \(\frac{1}{(4x^{2})^{2}}\). This step entirely removes the negative exponent.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and exponents combined according to the algebraic rules. Simplifying these expressions involves using exponent rules, such as the negative exponent rule and exponent of exponent rule, as well as understanding reciprocal relationships.

• Simplifying algebraic expressions is about reducing them to their simplest form while maintaining equality.
• It's a crucial part of algebra that allows for easier manipulation and solution of equations.

In our example, simplifying \((4x^{2})^{-2}\) required a clear understanding of algebraic concepts to correctly apply these rules and arrive at \(\frac{1}{16x^{4}}\). Mastering such techniques is key to succeeding in algebra and various other mathematical computations.