Problem 19 - Algebra for College Students

Question

Simplify each numerical expression. \(10^{-1} \cdot 10^{-2}\)

Step-by-Step Solution

Answer

\(10^{-3}\)

Step 1: Understand the Problem

We need to simplify the expression \(10^{-1} \cdot 10^{-2}\). It involves multiplying two numbers with the same base but different negative exponents.

Step 2: Recall the Properties of Exponents

Remember that when multiplying numbers with the same base, you keep the base and add the exponents: \(a^m \times a^n = a^{m+n}\).

Step 3: Apply the Properties of Exponents

Apply the property to the expression: \(10^{-1} \cdot 10^{-2} = 10^{-1 + (-2)}\).

Step 4: Simplify the Exponents

Add the exponents: \(-1 + (-2) = -3\). So the expression simplifies to \(10^{-3}\).

Step 5: Final Answer

The simplified form of the expression is \(10^{-3}\).

Key Concepts

Negative Exponents

Negative exponents can seem puzzling at first, but they're quite simple once you know the rule. When we have a number with a negative exponent, such as \(10^{-2}\), it means that we are dealing with the reciprocal of the base raised to the positive of that exponent. It's like saying, "Instead of multiplying, we're dividing." Let's break it down further:

• For example, \(a^{-n} = \frac{1}{a^n}\). You flip the number to the bottom of a fraction and make the exponent positive.
• So, \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\).

Remember that any base with a negative exponent can be converted to a fraction. Keep in mind that calculating with negative exponents is often about understanding fractions and dividing, rather than multiplication.

Multiplying Powers

When multiplying powers with the same base, you can simplify the process using an important rule of exponents. This rule states that you keep the base and add the exponents together. Here's how it works:

• If you have \(a^m \times a^n\), the result is \(a^{m+n}\).
• In our example, \(10^{-1} \times 10^{-2}\) simplifies to \(10^{-1 + (-2)}\).
• The calculation becomes \(-1 + (-2) = -3\), which means our expression simplifies to \(10^{-3}\).

By adding the exponents, we've reduced a potentially complicated multiplication into a simple addition of integers. This is a crucial step in simplifying expressions with exponents efficiently.

Properties of Exponents

Exponents have several properties that simplify expressions involving powers, and understanding these properties is key to mastering exponents. Here are a few fundamental properties:

• Product of Powers: \(a^m \times a^n = a^{m+n}\) allows us to add the exponents.
• Quotient of Powers: \(a^m \div a^n = a^{m-n}\), which helps when dividing powers.
• Power of a Power: \((a^m)^n = a^{m \times n}\), which is helpful when raising a power to another power.

These properties make working with exponents much simpler. They transform multiplication and division of powers into straightforward addition and subtraction of exponents. As seen in our original exercise, these properties allowed us to quickly and accurately simplify the expression \(10^{-1} \cdot 10^{-2}\) to \(10^{-3}\). Mastering these rules can save you lots of time and prevent errors in calculations.