Problem 15 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \left(\frac{1}{4}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression \( \left( \frac{1}{4} \right)^{-2} \) evaluates to 16.

Step 1: Understand Negative Exponents

A negative exponent signifies that the base should be taken as the reciprocal raised to the corresponding positive exponent. For any nonzero number \(a\), \(a^{-n} = \frac{1}{a^n}\).

Step 2: Convert the Negative Exponent

Given the expression \( \left( \frac{1}{4} \right)^{-2} \), we convert it using the rule for negative exponents: \( \left( \frac{1}{4} \right)^{-2} = \left( \frac{4}{1} \right)^2 \).

Step 3: Simplify the Expression

Now simplify \( \left( \frac{4}{1} \right)^2 = 4^2 \). This means you multiply 4 by itself: \(4 \times 4\).

Step 4: Calculate the Power

Calculate \(4^2\): \(4 \times 4 = 16\).

Key Concepts

Reciprocal

In mathematics, the reciprocal of a number is essentially one divided by that number. It is a fundamental concept that plays a key role in simplifying expressions, especially when dealing with negative exponents.
When you encounter an expression like \( \left( \frac{1}{4} \right)^{-2} \), a negative exponent indicates that you should take the reciprocal of the base and then raise it to the positive version of the exponent.
For example:

• The reciprocal of \( \frac{1}{4} \) is \( 4 \), since \( \frac{1}{4} \times 4 = 1 \).
• Applying this, you have \( \left( \frac{1}{4} \right)^{-2} = \left( 4 \right)^2 \).

Understanding reciprocals will greatly help in managing expressions and simplifying calculations.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing an expression to its simplest form. This allows for easier computation and understanding of complex problems.
When working with negative exponents, it is particularly important to transition towards the expression's reciprocal before simplifying.
Let's look at the expression \( \left( 4 \right)^2 \). This can be simplified by:

• Identifying the base: Here, the base is \( 4 \).
• Applying the exponent: Multiply the base by itself as many times as the exponent indicates. Hence, \( 4 \times 4 \).
• Arriving at the simplest form: In this case, \( 16 \).

Following these steps ensures that expressions are simplified correctly and efficiently.

Powers and Exponents

Powers and exponents are mathematical notations used to express repeated multiplication of the same number. Let's break this down: an exponent tells you how many times to multiply the base by itself.
In \( 4^2 \), the number \( 4 \) is the base, and \( 2 \) is the exponent. It means you need to multiply \( 4 \) two times, or \( 4 \times 4 \).
Here are some key points to remember:

• Positive exponents: Indicate regular multiplication. For example, \( 3^2 = 3 \times 3 = 9 \).
• Negative exponents: Indicate that you should find the reciprocal of the base. Hence, for any nonzero base \( a \), \( a^{-n} = \frac{1}{a^n} \).
• Zero as an exponent: Any non-zero number raised to the power of zero equals \( 1 \). For example, \( 5^0 = 1 \).

Mastering the use of powers and exponents will empower students to solve more complex equations with confidence.