Problem 36 - Algebra for College Students

Question

Simplify each numerical expression. \(\frac{10^{-2}}{10^{-5}}\)

Step-by-Step Solution

Answer

The simplified expression is 1000.

Step 1: Understanding Exponents

The problem involves dividing numbers in exponential format. Here, \(10^{-2}\) is the numerator and \(10^{-5}\) is the denominator.

Step 2: Apply the Quotient of Powers Rule

When dividing like bases with exponents, apply the rule: \(\frac{a^m}{a^n} = a^{m-n}\). In this case, simplify by calculating the exponents: \(10^{-2 - (-5)}\).

Step 3: Calculate the Exponent

Subtract the exponents: \(-2 - (-5) = -2 + 5 = 3\).

Step 4: Final Simplification

The expression reduces to: \(10^3\). Simplify further by calculating the power: \(10^3 = 1000\).

Key Concepts

Quotient of Powers Rule

When working with exponents, there's a helpful rule known as the "Quotient of Powers Rule". This rule makes it easier to simplify expressions where you are dividing two exponential terms with the same base. The rule states: \( \frac{a^m}{a^n} = a^{m-n} \). In plain terms, when you divide numbers with the same base, you subtract the exponents from each other.

For example, consider the expression \( \frac{10^{-2}}{10^{-5}} \). Here, both terms have a base of 10, making it a perfect candidate for the quotient of powers rule. By applying the rule, you subtract the exponent of the denominator (-5) from the exponent of the numerator (-2), resulting in: \(-2 - (-5) = -2 + 5 = 3\). Thus, the expression simplifies to \( 10^3 \). "How do I use it in other cases?" you might ask. Simply ensure the bases are the same, then subtract the exponents. If the result requires further calculation, proceed with that to get your final answer.

Exponential Format

Exponents are a part of the exponential format, a mathematical notation that expresses numbers as powers. For example, \(10^3\) means 10 is multiplied by itself three times: \(10 \times 10 \times 10\). This format is especially useful for expressing very large or very small numbers simply.

Exponential expressions are identified by a base and an exponent. Here, the base is 10, indicating the number being multiplied. The exponent (3 in our example) tells us how many times the base is multiplied by itself.

In our original exercise, \(10^{-2}\) and \(10^{-5}\) are both in exponential format. The negative exponents indicate division or reciprocal. So \(10^{-2}\) is \(\frac{1}{10^2} = \frac{1}{100}\). Understanding these components is crucial when simplifying expressions, as it allows easier manipulation and calculation.

Numerical Expression Simplification

Numerical expression simplification involves reducing a complex expression to its simplest form. In our example exercise, this is the final step where we move from \( \frac{10^{-2}}{10^{-5}} \) to \(10^3\), and then finally to 1000.

Let's break this down with key steps:

• Identify the terms and their bases. Check if you can employ rules such as the quotient of powers rule.
• Perform any needed operations on the exponents (subtract, in our case).
• Compute the final numerical answer. For \(10^3\), calculate: \(10 \times 10 \times 10 = 1000\).

By following these steps, simplification becomes straightforward. Remember, the practice of consistently breaking down expressions ensures you have a clear path to the simplified form.