Problem 63 - Elementary Algebra

Question

Perform the following operations. $$ \left(4 \times 10^{2}\right)\left(8 \times 10^{6}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given expression in scientific notation: (4 × 10²)(8 × 10⁶) Answer: 32 × 10⁸

Step 1: Multiply the coefficients

We will first multiply the coefficients, which are 4 and 8. $$ 4 \times 8 = 32 $$

Step 2: Add the exponents

Next, we will add the exponents from the powers of 10. The exponents are 2 and 6. $$ 2 + 6 = 8 $$

Step 3: Combine the results

Finally, we will combine the results from step 1 and step 2 by multiplying the result from step 1 with 10 raised to the power of the result from step 2. $$ 32 \times 10^{8} $$ Thus, the result of the given operation is: $$ \left(4 \times 10^{2}\right)\left(8 \times 10^{6}\right) = 32 \times 10^{8} $$

Key Concepts

Multiplying Exponents

Multiplying exponents might seem tricky at first, but once you get the hang of it, it becomes much more straightforward. When dealing with expressions where the base is the same and you're multiplying them, there's a simple rule you can apply. Courtesy of the properties of exponents, you only need to add the exponents instead of multiplying them. This rule stems from the fact that exponents are shortcuts for repeated multiplication.

For example, consider the expression \( (10^2)(10^6) \). Instead of multiplying out each term, which would take a while, just add the exponents: 2 and 6. This gives you \( 10^{2+6} = 10^8 \).

By adding the exponents, you've simplified a potentially complex multiplication into something much quicker and easier! This concept is handy across mathematics and science, especially when dealing with scientific notation.

Powers of Ten

The powers of ten are both simple and powerful tools in mathematics. These are numbers like \( 10^1 = 10 \), \( 10^2 = 100 \), \( 10^3 = 1000 \), and so on. Each power increases by just one zero after the previous power.

Why is this important? Because multiplying by powers of ten is incredibly efficient. You don't actually perform multiplication as you often would. Instead, you shift the decimal point in a number to the right for positive exponents, or to the left for negative exponents.

For example, \( 3.5 \times 10^2 \) becomes 350, simply by moving the decimal two places to the right. Similarly, \( 3.5 \times 10^{-2} \) equals 0.035, moving the decimal two places left.

Powers of ten simplify calculations and help manage large and small numbers by keeping them concise and easy to read, which is especially crucial in scientific notation.

Coefficients in Scientific Notation

Coefficients are key elements in scientific notation, providing the significant figures part of a number. In a term like \( 4 \times 10^2 \), 4 is the coefficient. It represents the important non-zero digits of the number in scientific terms.

When working with scientific notation, you often multiply these coefficients together, just like any regular numbers. For example, consider \( (4 \times 10^2)(8 \times 10^6) \). Multiply the coefficients (4 and 8) first, giving you 32.

Scientific notation is a powerful tool because of its use of coefficients. It allows us to express extremely large or small numbers succinctly, while the powers of ten handle the rest, indicating how far the decimal point has moved. By manipulating the coefficient and keeping the powers of ten in check, scientists and mathematicians can easily communicate measurements, calculations, and complex equations.