Problem 30 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 10^{-5} \cdot 10^{7} $$

Step-by-Step Solution

Answer

\(\ 10^{2} \) which equals to 100.

Step 1: Identify the rule

The rule to use here is the multiplication rule of exponents, which states that for any positive integer \( a \), \( a^n \cdot a^m = a^{n+m} \).

Step 2: Apply the rule

Now apply this rule to the given equation. Add the exponents of \( 10^{-5} \) and \( 10^{7} \). This gives us \( 10^{-5+7} \).

Step 3: Simplify the equation

Next, simplify the exponent. \( -5 + 7 = 2 \). Thus, the equation simplifies to \( 10^{2} \).

Key Concepts

Multiplication Rule of Exponents

The multiplication rule of exponents is a fundamental concept in mathematics that simplifies working with powers. When you multiply two expressions that have the same base, you can combine them using this rule. The rule states that for any base \(a\) and exponents \(n\) and \(m\), the expression \(a^n \cdot a^m\) simplifies to \(a^{n+m}\). It essentially tells us that we add the exponents when multiplying like bases. This rule only applies when the bases are identical.

In our exercise, the base is \(10\), which remains consistent in both parts of the expression: \(10^{-5} \cdot 10^{7}\). Using the multiplication rule, we simply add the exponents \(-5\) and \(7\) to simplify the expression to \(10^{2}\). This powerful rule helps streamline complex mathematical operations involving exponents, making it easier to evaluate expressions.

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical expressions into their simplest form. By employing specific rules and strategies, we can make expressions easier to evaluate and interpret. One of the tools for simplifying expressions with exponents is the multiplication rule of exponents.

In the provided exercise, we start with the expression \(10^{-5} \cdot 10^{7}\). After applying the multiplication rule of exponents, we obtained \(10^{-5+7}\). The next step is to simplify the exponent by performing the operation \(-5 + 7\).

Thus, the problem reduces to finding the sum of the exponents: \(2\). Now, our expression becomes \(10^2\). <

By reducing the expression to its simplest form, it becomes more convenient for numerical evaluation and demonstrates the effectiveness of employing exponent rules in algebraic simplification.

Integer Exponents

Understanding integer exponents is essential in simplifying and evaluating expressions, especially those involving powers. Exponents denote how many times a base is multiplied by itself. When exponents are integers, which include negative integers, they express repeated multiplication or division.

In an expression like \(10^{-5}\), the negative exponent \(-5\) implies a reciprocal. It means \(10^{-5} = \frac{1}{10^5}\), hence involving division rather than multiplication. Conversely, positive exponents correspond to straightforward repeated multiplication.

When working with integer exponents, it's important to accurately apply operations like addition and subtraction, especially when simplifying or combining terms.

The initial exercise offered exponents \(-5\) and \(7\), both integers, which were combined and simplified to \(2\) using the multiplication rule of exponents. By mastering integer exponents, problems that would otherwise seem daunting become manageable and can be resolved with ease.