Problem 109 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ 4 \cdot 9^{-2} $$

Step-by-Step Solution

Answer

Based on the given step-by-step solution, answer the following question: Question: Evaluate the numerical expression \(4 \cdot 9^{-2}\). Answer: \(\frac{4}{81}\)

Step 1: Evaluate the exponent

We need to simplify the exponent, \(9^{-2}\), first according to the order of operations. To evaluate it, recall that a negative exponent means taking the reciprocal of the base and changing the exponent's sign. So, \(9^{-2}\) is the same as \(\frac{1}{9^2}\).

Step 2: Simplify the exponent

Now, we need to calculate \(9^2\). Since \(9^2\) means \(9 \cdot 9\), we have \(9^2 = 81\). Thus, \(9^{-2} = \frac{1}{9^2} = \frac{1}{81}\).

Step 3: Perform the multiplication

Now, we can evaluate the expression: \(4 \cdot 9^{-2}\). Substitute \(9^{-2}\) with \(\frac{1}{81}\): $$4\cdot \frac{1}{81}$$

Step 4: Simplify the expression

To simplify this expression, multiply 4 by the numerator (1) and keep the denominator (81) the same: $$4\cdot \frac{1}{81} = \frac{4\cdot 1}{81} = \frac{4}{81}$$ The evaluated numerical expression is \(\frac{4}{81}\).

Key Concepts

Order of Operations

When it comes to solving mathematical expressions, the order of operations is like a set of rules that you need to follow to get the correct answer. This is usually remembered by the acronym PEMDAS which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the problem we are discussing, first, we had to address the exponent before multiplication. This ensures that we follow the rules and avoid any errors in our calculations. Evaluating exponents first, like the problem's exponent of \(9^{-2}\), is crucial before moving on to other operations.

Negative Exponents

Negative exponents might seem a little tricky at first, but they actually follow a straightforward rule. When you have a base with a negative exponent, you take the reciprocal of the base and change the sign of the exponent to positive.
For example, with \(9^{-2}\), instead of thinking about multiplying \(9\) by itself, we think of taking \(\frac{1}{9^2}\). This is because a negative exponent flips the fraction from being part of the numerator to the denominator. Remember, changing negative exponents to positive ones by flipping them into fractions makes them much easier to deal with!

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form. Often, this involves making sure you are working with the lowest possible numbers making calculations easier.
In the example, you already started with \(\frac{1}{81}\) when addressing \(9^{-2}\). Trying to simplify these fractions further isn't always necessary, but understanding that we keep the fraction as-is ensures correct multiplication later. It's important to get comfortable with this process so you can simplify or multiply fractions quickly and accurately when needed.

Multiplication of Fractions

The multiplication of fractions is as easy as multiplying the numerators together and the denominators together. Once you have them set up, the process is straightforward.For instance, in solving \(4 \cdot \frac{1}{81}\), you simply multiply \(4\) by \(1\) (the numerator) and keep \(81\) (the denominator) the same, giving you \(\frac{4}{81}\).
Keeping track of the numbers during multiplication is essential as fractions might seem different from whole numbers, but with simple steps, they become manageable. Always remember to simplify further if possible, though in this case, \(\frac{4}{81}\) is as simplified as it gets.