Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write each expression as a rational number without an exponent. $$ 4^{-2} $$

Step-by-Step Solution

Answer

The expression \(4^{-2}\) is equal to \(\frac{1}{16}\) as a rational number.

Step 1: Understand Negative Exponents

A negative exponent indicates that the base is on the wrong side of the fraction line. In other words, you move the base to the other side to make the exponent positive. If you have a term like \(a^{-b}\), it becomes \(\frac{1}{a^b}\) when expressed as a positive exponent.

Step 2: Apply Rule to Given Expression

Given the expression \(4^{-2}\), apply the rule for negative exponents. This means moving 4 to the denominator and changing the exponent to a positive. This results in: \[4^{-2} = \frac{1}{4^2}\].

Step 3: Simplify the Denominator

Now, calculate \(4^2\) in the denominator. \(4^2\) means 4 multiplied by itself, which is 16. So, \(\frac{1}{4^2} = \frac{1}{16}\).

Step 4: Express the Answer as a Rational Number

The given expression \(4^{-2}\) can now be written as the rational number \(\frac{1}{16}\) without any exponent.

Key Concepts

Rational Numbers

Rational numbers are simply numbers that can be expressed as a fraction, where both the numerator and the denominator are integers. The denominator in these fractions cannot be zero. Rational numbers include positive numbers, negative numbers, and the number zero. For instance, common examples include \( \frac{1}{2} \), \( \frac{-3}{4} \), and \( 0 \). Any integer is also a rational number because it can be expressed as a fraction with a denominator of one, like \( \frac{5}{1} \) for the integer 5.

In the context of our exercise, when we convert expressions with negative exponents like \( 4^{-2} \), the goal is to express it as a rational number. We do this by converting the expression into a fraction. In our case, \( 4^{-2} \) becomes \( \frac{1}{16} \), which is a straightforward rational number format. This helps to simplify mathematical expressions and make them manageable and useful for further computation.

Exponent Rules

The rules of exponents are crucial in simplifying expressions and converting them into different forms. Let's focus on a few key rules that are particularly relevant:

• Negative Exponents: When you see a negative exponent, it means you'll take the reciprocal of the base raised to the positive of that exponent. For example, \( a^{-b} \) becomes \( \frac{1}{a^b} \).
• Zero Exponent Rule: Any non-zero base raised to the power of zero is 1. For instance, \( 5^0 = 1 \).
• Power of a Power: When you raise a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \cdot n} \).

Applying these rules helps in rearranging and simplifying mathematical expressions. In our example, the negative exponent rule helps us convert \( 4^{-2} \) to its reciprocal form, \( \frac{1}{4^2} \), ultimately simplifying it to \( \frac{1}{16} \). The simplification using exponent rules can make calculations more straightforward and results less complex.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with. This often requires using properties of numbers and operations, including exponent rules. Let's break it down:

First, identify any components such as exponents or operations that can be simplified. For example, in \( 4^{-2} \), the negative exponent prompts us to use the negative exponent rule.

Next, apply the necessary rule to simplify the expression. For \( 4^{-2} \), we use the rule \( a^{-b} = \frac{1}{a^b} \), transforming it to \( \frac{1}{4^2} \).

Finally, compute any calculations to finish simplifying. Here, \( 4^2 \) is simplified to 16, so the expression \( \frac{1}{4^2} \) becomes \( \frac{1}{16} \).

Simplification helps to increase clarity and make expressions ready for further mathematical operations. It's an essential step, especially in more complex algebraic scenarios. The outcome is a clarified, manageable, and ready-to-use rational number. Understanding each step and the rules applied allows for smooth problem-solving transitions.