Problem 39 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ 10^{0} \cdot 10^{6} \cdot 10^{2} $$

Step-by-Step Solution

Answer

The simplified form is \( 10^{8} \).

Step 1: Understanding the Product Rule

The product rule for exponents states that when multiplying two expressions with the same base, you add the exponents. Mathematically, this is expressed as \( a^{m} \cdot a^{n} = a^{m+n} \). Here, the base is 10.

Step 2: Applying the Product Rule

We have \( 10^{0} \cdot 10^{6} \cdot 10^{2} \). Using the product rule, we add the exponents: \( 0 + 6 + 2 \).

Step 3: Calculating the Sum of the Exponents

Add the exponents together: \( 0 + 6 + 2 = 8 \).

Step 4: Express the Simplified Form

Now that we have the sum of the exponents, we express the simplified form using the base 10: \( 10^{8} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra and calculus. The goal is to reduce a complex expression to its simplest form for easier understanding and further calculations. Often, simplifying involves using mathematical rules or properties to combine like terms or remove unnecessary parts.

When working with expressions involving exponents, applying the correct rules can help simplify them effectively:

• First, identify like terms. In expressions involving exponential terms, like terms have the same base.
• Use rules like the product rule for exponents to combine these terms into a simpler expression.
• Always check your work to ensure that your simpler expression is equivalent to the original.

This process not only streamlines mathematical work but also builds a deeper understanding of how numbers interact.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, referred to as the exponent. It is a shorthand method of expressing repeated multiplication.

For example:

• The expression \( 10^3 \) means multiplying 10 by itself three times, which equals 1000.
• In general, \( a^n \) means the number \( a \) is multiplied by itself \( n \) times.

Understanding exponentiation is vital because it forms the foundation for more advanced topics in mathematics such as logarithms and exponential growth.

When dealing with exponents, always remember that changing either the base or the exponent affects the overall value significantly.
Also, special cases like zero and negative exponents have unique rules in mathematics.

Properties of Exponents

The properties of exponents, also referred to as laws of exponents, are a set of rules that describe how to handle expressions involving powers. These properties help in simplifying expressions and solving exponential equations.

Some important properties include:

• Product Rule: When multiplying like bases, add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).

By understanding these properties, one can easily manipulate and simplify expressions involving exponents.
Being comfortable with these rules will significantly ease calculations and problem-solving in both basic and advanced mathematics.