Problem 18 - Algebra for College Students

Question

Simplify each numerical expression. \(10^{4} \cdot 10^{-6}\)

Step-by-Step Solution

Answer

The simplified form is \(10^{-2}\) or \(\frac{1}{100}\).

Step 1: Identify the Rule of Exponents

To simplify the expression, recognize that you are multiplying numbers with the same base, which allows the use of the Product of Powers Rule. The rule states that when you multiply like bases, you can add their exponents: \(a^m \cdot a^n = a^{m+n}\).

Step 2: Apply the Product of Powers Rule

Using the rule from Step 1, add the exponents together for the base 10: \(10^{4} \cdot 10^{-6} = 10^{4 + (-6)}\). This simplifies to \(10^{-2}\).

Step 3: Simplify Further if Necessary

Since the expression \(10^{-2}\) is already simplified as an exponential expression, you may also express it as a fraction if needed: \(10^{-2} = \frac{1}{10^2}\), which further simplifies to \(\frac{1}{100}\).

Key Concepts

Product of Powers Rule

When working with exponents, especially when multiplying terms with the same base, the Product of Powers Rule is a very handy tool. This rule lets you simplify expressions efficiently. Here’s how it works: when two exponents with the same base are multiplied, you simply add their exponents together.

For example, if you have an expression such as \(a^m \cdot a^n\), the Product of Powers Rule tells us that this is equivalent to \(a^{m+n}\). This means we take the base \(a\) and add the exponents \(m\) and \(n\) together. This rule simplifies otherwise complex multiplications, making calculations much easier.

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill for solving mathematical problems. It involves performing operations to present the expression in its simplest form.

In the context of exponents, simplification often involves applying rules like the Product of Powers Rule. For instance, with the expression \(10^4 \cdot 10^{-6}\), by simply recognizing the common base (10), you can apply the Product of Powers Rule, which tells us to add the exponents: \(4 + (-6) = -2\). This results in \(10^{-2}\), which is simpler than the original expression.

Simplifying expressions not only makes equations more manageable but also reveals insights into the properties of numbers and their relationships.

Negative Exponents

Negative exponents might appear confusing at first, but they simply represent the reciprocal of the base raised to the positive version of the given power. In other words, if you have an expression \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). This is a useful concept because it allows you to convert large expressions into more manageable forms without difficulty.

In the example of \(10^{-2}\), applying the definition of negative exponents, you can convert this to \(\frac{1}{10^2}\), further simplifying to \(\frac{1}{100}\). Knowing how to work with negative exponents is crucial for solving many algebraic problems, especially those involving fractions and division of powers. By understanding negative exponents, you open the door to a wide array of mathematical concepts and applications.