Problem 9 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(10^{2} \cdot 10^{4}\)

Step-by-Step Solution

Answer

The simplified expression is \(10^6\).

Step 1: Identify the powers of 10

In the expression \(10^2 \cdot 10^4 \), identify the powers involved, which are 2 and 4.

Step 2: Apply the product of powers property

Use the property that states when multiplying two powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). Here the base is 10, so we add the exponents: \(2 + 4\).

Step 3: Simplify the expression

Add the exponents: \(2 + 4 = 6\). Thus, \(10^2 \cdot 10^4 = 10^6\).

Key Concepts

Understanding Powers of 10

When we talk about powers of 10, we're referring to numbers that have 10 as a base with an exponent. The exponent tells us how many times to multiply the base number (in this case, 10) by itself. This is a cornerstone of scientific notation and many mathematical concepts as it helps break down and simplify very large or very small numbers.

• A positive exponent indicates how many times 10 is multiplied by itself. For example, in the expression \(10^2\), it means \(10 \times 10\), which equals 100.
• As the exponent increases, the value of the number becomes larger exponentially. For instance, \(10^4\) equals 10,000 because you multiply four 10s together: \(10 \times 10 \times 10 \times 10\).

Powers of 10 are particularly simple to interpret because they correspond directly to the number of zeros following the 1 when the number is written in standard form. This simplicity makes them a powerful tool in calculations and equations.

The Product of Powers Property Explained

The product of powers property is a handy rule in mathematics that simplifies multiplying expressions with the same base. When you have two powers with the same base, this property allows you to simply add the exponents together instead of multiplying out each power separately.
Let's break it down with the exercise example: - Consider the expression \(10^2 \cdot 10^4\).- According to the product of powers property, \(a^m \cdot a^n = a^{m+n}\). This means that since both terms have the same base (10), we can add the exponents: \(2 + 4\).
So instead of calculating \(10 \times 10\) and then \(10 \times 10 \times 10 \times 10\) separately, we've condensed all the multiplying into one easy step. This is how we arrive at \(10^6\) directly and efficiently. This property not only saves time but simplifies expressions significantly in algebra and calculus.

Steps to Simplifying Expressions

Simplifying mathematical expressions, especially those involving exponents, is a crucial skill in algebra. It involves reducing the expression to its simplest form, making it easier to read and work with.
When simplifying an expression like \(10^2 \cdot 10^4\), the steps are straightforward:

• Identify the base and exponents: Recognize that both terms share a common base (10 in this case) and note the respective exponents (2 and 4).
• Apply the product of powers property: Add the exponents together because the base is the same. In this case, \(2 + 4 = 6\).
• Write the simplified expression: The expression \(10^2 \cdot 10^4\) simplifies to \(10^6\).

By following these steps, students can always ensure their answers are as clear and simple as possible, paving the way for solving more complex equations with confidence.