Problem 18 - Elementary Algebra

Question

Perform each multiplication. $$ \left(1 \times 10^{-4}\right)\left(6 \times 10^{24}\right) $$

Step-by-Step Solution

Answer

Answer: \(6 \times 10^{20}\)

Step 1: Multiply the coefficients

Firstly, we need to multiply the coefficients of the two numbers: \(1\) and \(6\). The result is: $$ 1 \times 6 = 6 $$

Step 2: Multiply the powers of 10

Now, we need to multiply the powers of \(10\), which are \(10^{-4}\) and \(10^{24}\). We can do this by simply adding the exponents, following the rules for multiplying exponential terms with the same base: $$ 10^{-4} \times 10^{24} = 10^{(-4 + 24)} = 10^{20} $$

Step 3: Combine the results

Finally, we combine the results from Steps 1 and 2 to get the answer: $$ (1 \times 10^{-4}) (6 \times 10^{24}) = 6 \times 10^{20} $$ The result of the multiplication is \(6 \times 10^{20}\).

Key Concepts

Multiplying Exponents

Understanding how to multiply exponents is an essential skill in mathematics, especially when dealing with scientific notation. When you encounter terms like \(10^{-4} \times 10^{24}\), you are working with exponential terms that have the same base (in this case, base 10). To multiply these terms, you simply add the exponents together. The general rule for multiplying exponents with the same base is:

• Base remains the same.
• Add the exponents together.

For example, for \(a^m \times a^n\), the result is \(a^{m+n}\). Thus, when multiplying \(10^{-4}\) and \(10^{24}\), we add \(-4\) and \(24\), giving us \(10^{20}\). This rule simplifies calculations and is particularly useful for multiplying exponential terms in scientific notation.

Exponential Terms

Exponential terms are numbers expressed in the form of a constant base raised to an exponent. This format is fundamental in scientific notation, which is used to represent very large or very small numbers concisely. The base is usually 10, especially in scientific contexts. Exponential terms such as \(10^n\) represent the number 10 raised to the power \(n\). The exponent \(n\) indicates how many times the base is multiplied by itself. For instance:

• \(10^2 = 10 \times 10 = 100\)
• \(10^{-1} = \frac{1}{10} = 0.1\)

Whether dealing with positive or negative exponents, understanding how they work allows for quick computation and comprehension of vast numerical scales, such as distances in astronomy or microscopic sizes in biology.

Powers of Ten

Powers of ten play a crucial role in scientific notation and make it easier to handle extremely large or small numbers. Each power of ten varies by factors of ten, moving up or down the scale:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1,000\)
• ... and so on.

Conversely, negative powers of ten represent fractional numbers:

• \(10^{-1} = 0.1\)
• \(10^{-2} = 0.01\)
• ... and continue as decimals get smaller.

Understanding powers of ten simplifies many mathematical operations. For example, multiplying \(6 \times 10^{24}\) by \(10^{-4}\) is straightforward by adjusting the exponents and provides a clear solution like \(6 \times 10^{20}\). This understanding is pivotal in scientific disciplines where quick and accurate estimations with large datasets are often required.