Problem 33 - Algebra for College Students

Question

For Problems \(33-50\), use scientific notation and the properties of exponents to help you perform the following operations. \((0.0037)(0.00002)\)

Step-by-Step Solution

Answer

The product is \(7.4 \times 10^{-8}\).

Step 1: Convert decimal to scientific notation

The first step is to convert each of the given numbers to scientific notation. The number 0.0037 can be rewritten as \(3.7 \times 10^{-3}\), and the number 0.00002 can be rewritten as \(2 \times 10^{-5}\).

Step 2: Multiply the constants

Multiply the constant parts of the scientific notations: \(3.7 \times 2 = 7.4\).

Step 3: Add the exponents

When multiplying numbers in scientific notation, we add the exponents of 10. The exponents from our numbers are \(-3\) and \(-5\). So, add these together: \(-3) + (-5) = -8\).

Step 4: Write the result in scientific notation

Combine the result from Step 2 and Step 3 to write the final answer in scientific notation. Therefore, the product is \(7.4 \times 10^{-8}\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential in mathematical calculations, especially when dealing with scientific notation. Exponents indicate how many times a number, known as the base, is multiplied by itself. For example, in scientific notation, exponents are used to express large or small numbers efficiently.

• When multiplying numbers with the same base, add their exponents. For example, for numbers of the form \( a^m \times a^n \), the result is \( a^{m+n} \).
• When dividing numbers with the same base, subtract their exponents. So, \( \frac{a^m}{a^n} = a^{m-n} \).
• An exponent of zero means the number equals 1 (\( a^0 = 1 \)).
• Negative exponents indicate a reciprocal, meaning \( a^{-n} = \frac{1}{a^n} \).

By applying these properties, calculations involving multiplication or division of numbers in scientific notation become straightforward. In this exercise, you multiply two numbers in scientific notation by multiplying their constant parts and adding the exponents from their powers of ten.

Multiplying Decimals

Multiplying decimals can initially seem confusing, but it's an extension of simple multiplication with some rules to account for decimal places. Here's a simplified approach:

• Ignore the decimal points and multiply the numbers as if they were whole numbers.
• Count the total number of decimal places in both of the numbers you are multiplying. This determines how many decimal places your final product should have.
• Place the decimal in the resulting product so that it has the same total number of decimal places counted in the step above.

For instance, when multiplying \(0.0037\) by \(0.00002\), first view them as \(37\) and \(2\) and conduct multiplication for a preliminary product of \(74\). Since there are five decimal places in total (three from \(0.0037\) and two from \(0.00002\)), your final answer in decimal form should be \(0.000074\). However, scientific notation, as shown, often simplifies this.

Conversion to Scientific Notation

Converting a number to scientific notation involves expressing it as the product of a number (between 1 and 10) and a power of 10. This notation simplifies handling very large or small numbers.

• Identify the appropriate decimal point shift needed to make the number between 1 and 10.
• Count the number of places the decimal point is moved. This count becomes the exponent of 10.
• If the decimal is moved to the left, the exponent is positive. If it is moved to the right, the exponent is negative.

Take \(0.0037\). Move the decimal three places to the right to rewrite it as \(3.7 \times 10^{-3}\). For \(0.00002\), shift the decimal five places to the right to get \(2 \times 10^{-5}\). This transformation is vital since it changes how numbers are handled in calculations, reducing errors associated with numerous decimal places and simplifying multiplication by focusing on the coefficients and exponents.