Problem 14 - Prealgebra

Question

Find each product or quotient. Express using exponents. $$10^{4} \cdot 10^{3}$$

Step-by-Step Solution

Answer

\(10^4 \cdot 10^3 = 10^7\)

Step 1: Understand the Exponent Rule for Multiplication

When multiplying two powers with the same base, we can add their exponents together. This is the product of powers property: \[a^m \cdot a^n = a^{m+n}\] where \(a\) is the base, and \(m\), \(n\) are the exponents.

Step 2: Apply the Rule to the Expression

Given \(10^4 \cdot 10^3\), both terms have the same base of 10. Using the product of powers property, add the exponents: \[10^4 \cdot 10^3 = 10^{4+3}\]

Step 3: Simplify the Exponents

Simplify \(4 + 3\) to get \(7\). Therefore, the expression becomes \(10^7\).

Key Concepts

Product of Powers Property

The Product of Powers Property is a foundational rule in understanding how to handle expressions involving exponents, especially during multiplication. This property is super helpful because it simplifies expressions so you don't have to manually multiply large numbers.

When you multiply two powers with the same base, you simply add their exponents. This is written as:

• If you have an expression like \[a^m \cdot a^n\], you can rewrite it as \[a^{m+n}\].

The letter "a" represents any number (the base), and "m" and "n" are the exponents that you add together.

For example, in our exercise \[10^4 \cdot 10^3\], both powers have the base 10. Using the product of powers, you add the exponents: \[4 + 3\] to get \[10^{7}\].

This property makes calculations much easier and is an essential tool in algebra.

Multiplication of Exponents

Multiplying numbers with exponents can be intimidating at first, but it becomes straightforward once you grasp the process. The trick lies in focusing on the base numbers and how the exponents are managed during multiplication.

When you have the same base, multiplication becomes a process of adding the exponents, thanks to the product of powers property, which simplifies your work remarkably.

Let's illustrate with an example:

• Consider \[10^4 \cdot 10^3\]. Both numbers 10, in this case, are multiplied together as one base number raised to a power.
• Instead of multiplying 10 by itself several times, you use the product of powers rule to add the exponents: \[4 + 3\], making it \[10^7\].

This outcome shows how multiplication of exponents simplifies complex arithmetic into more manageable calculations.

It's crucial to remember to only add exponents when bases are the same. This technique not only saves time but also makes error checking simpler.

Exponent Rules

Exponent rules are the guidelines that help us manipulate expressions involving powers and make complicated math simpler. Grasping these rules can open up an array of shortcuts in mathematics.

Here are a few fundamental rules that govern how exponents work:

• **Product of Powers:** For two numbers with the same base, add their exponents: \[a^m \cdot a^n = a^{m+n}\].
• **Power of a Power:** When raising a power to another power, you multiply the exponents: \[(a^m)^n = a^{m\cdot n}\].
• **Zero Exponent Rule:** Any base (except zero) raised to the power of zero is 1, so \[a^0 = 1\].
• **Negative Exponent Rule:** A base with a negative exponent means taking the reciprocal: \[a^{-n} = \frac{1}{a^n}\].

By learning and applying these exponent rules, solving expressions with several powers becomes more intuitive. For the example \[10^4 \cdot 10^3\], we primarily used the product of powers rule to efficiently find the answer, showing the usefulness of these rules in practical settings.