Problem 41 - Technical Mathematics with Calculus

Question

Multiply the following powers of \(10 .\) $$10^{-5} \cdot 10^{-4}$$

Step-by-Step Solution

Answer

\(10^{-5} \times 10^{-4} = 10^{-9}\).

Step 1: Understanding the Multiplication of Powers with the Same Base

When multiplying powers that have the same base, you can add the exponents. The base in this case is 10, and the rule is expressed as: if you have two numbers of the form \(10^a \times 10^b\), you can combine them into a single term \(10^{a+b}\).

Step 2: Apply the Rule to the Given Numbers

For the given numbers \(10^{-5} \times 10^{-4}\), apply the rule of adding exponents because their bases are the same (10). The exponents are -5 and -4, so you add them together: \(-5 + (-4) = -9\).

Step 3: Write the Final Answer

After adding the exponents, the final answer is \(10^{-9}\). This is because the sum of the exponents is -9.

Key Concepts

Negative Exponents

When dealing with negative exponents, it's essential to understand that they are not the same as negative numbers. Instead, a negative exponent represents the reciprocal of the base raised to the positive of that exponent.

For example, the expression \(10^{-5}\) can be interpreted as \(\frac{1}{10^5}\) or one divided by ten thousand. Similarly, \(10^{-4}\) is equivalent to \(\frac{1}{10^4}\) or one divided by ten thousand.

To multiply numbers like \(10^{-5} \times 10^{-4}\), you follow the standard exponent rules but keep in mind that the resulting power is still in the form of a reciprocal. This leads us to another critical exponent rule which is adding exponents when the bases are the same.

Exponent Rules

Exponent rules, or laws of exponents, are fundamental to understanding arithmetic and algebraic operations involving powers. Particularly, when multiplying powers with the same base, you add the exponents together. This is succinctly represented as \(a^m \times a^n = a^{m+n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents.

In our problem, the base is 10, and when we apply this rule to the exponents \(-5\) and \(-4\), we get \(10^{-5} \times 10^{-4} = 10^{-5 + (-4)} = 10^{-9}\). It's vital to understand that the result of adding negative numbers will also be negative, so the exponent in the answer is negative. This compels us to express our final answer in another format that can be easier to understand and use, which leads us to scientific notation.

Scientific Notation

Scientific notation is a way to write very large or very small numbers clearly and concisely, often used in science and engineering to make calculations easier. Any number expressed in scientific notation is in the form of \(a \times 10^n\), where \(1 \<= a \< 10\) and \(n\) is an integer.

In the context of our problem, \(10^{-9}\) is already in a form of scientific notation, where \(a\) is 1, and \(n\) is -9. If we had a different number being multiplied by this power of 10, say, \(3.2 \times 10^{-5} \times 10^{-4}\), we would first add the exponents to get \(3.2 \times 10^{-9}\), keeping the consolidated scientific notation.

This notation is not only streamlined but also allows for easier comparison between very large or small numbers and can be useful when converting to and from standard notation for practical applications.