Problem 9 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (4 \cdot 1)^{-2} $$

Step-by-Step Solution

Answer

The short answer is \(\frac{1}{16}\).

Step 1: Simplify the expression within the parentheses

First, simplify the expression in the parentheses. This gives 4*1, which equals 4. So, the expression becomes \(4^{-2}\).

Step 2: Apply the property of negative exponents

Next, apply the property that a number raised to the power of -n is equivalent to the reciprocal of the number raised to the power of n. This gives the new expression as \(\frac{1}{4^2}\).

Step 3: Evaluate the denominator

Lastly, evaluate the denominator of the expression. The denominator is 4^2, which equals 16. This gives the final answer as \(\frac{1}{16}\).

Key Concepts

Simplifying Expressions

When it comes to simplifying expressions, the first step is to work inside the parentheses. The expression provided to us is \((4 \cdot 1)^{-2}\). To simplify what is inside the parentheses, calculate the product: \(4 \cdot 1 = 4\). This simplifies the expression to \(4^{-2}\). Inside parentheses, always perform operations like multiplication or division first to make further simplifications easier. Once simplified, you can focus on handling the negative exponents more effectively. Simplifying expressions is all about breaking them down into smaller, manageable parts.

Reciprocal

The concept of a reciprocal is key when dealing with negative exponents. If you see a negative exponent, it suggests using the reciprocal of the base. For instance, in our expression \(4^{-2}\), the negative exponent indicates a flip situation:

• The base \(4\) stays the same.
• The expression becomes \(\frac{1}{4^2}\) due to the reciprocal.

A reciprocal is simply the flipped version of the number. For any number \(a\), the reciprocal is \(\frac{1}{a}\). For example, the reciprocal of \(4\) is \(\frac{1}{4}\). Using this idea helps you convert negative exponents into positive ones, making computations simpler.

Exponent Rules

Understanding exponent rules is essential for simplifying and evaluating expressions like \(4^{-2}\). One fundamental rule is that a negative exponent, such as \(x^{-n}\), means you take the reciprocal of the base, \(x\), and apply the positive exponent: \(\frac{1}{x^n}\). Using this rule for \(4^{-2}\):

• First, convert it to \(\frac{1}{4^2}\).
• Then, calculate \(4^2\), which is \(16\).
• This gives you the final result of \(\frac{1}{16}\).

Exponent rules make it straightforward to handle both positive and negative exponents, allowing more complex expressions to be simplified and understood easily. Practice these conversions and use the rule consistently for better mastery of mathematical expressions.