Problem 69 - 1,001 GED Practice Questions For Dummies

Question

What is the product of \(\left(8 \times 10^6\right) \times\left(4 \times 10^{-3}\right)\) ? Write your answer in scientific notation. (Note: You may not use a calculator.) (A) \(3.2 \times 10^{-17}\) (B) \(3.2 \times 10^2\) (C) \(32 \times 10^3\) (D) \(3.2 \times 10^4\)

Step-by-Step Solution

Answer

(D) \(3.2 \times 10^4\)

Step 1: Multiply the base numbers

First, we need to multiply the base numbers (the numbers without exponents). In this case, the base numbers are 8 and 4. So, we have: \(8 \times 4 = 32\)

Step 2: Multiply the powers of 10

Next, we need to multiply the powers of 10. Recall the rule for multiplying numbers with the same base and different exponents which states - when multiplying, add the exponents together: \((10^6) \times (10^{-3}) = 10^{(6 + -3)} = 10^3\)

Step 3: Combine the products

Now, we can combine the results from step 1 (32) and step 2 (\(10^3\)) to get: \(32 \times 10^3\)

Step 4: Check the answer format

The product should be in scientific notation which means that there should be only one digit to the left of the decimal point. Our result, however, has two digits. So we need to modify it to fit the scientific notation format: \(32 \times 10^3 = 3.2 \times 10 \times 10^3 = 3.2 \times 10^4\)

Step 5: Choose the correct answer

Now, we can see that the correct answer is (D) \(3.2 \times 10^4\).

Key Concepts

Exponent Rules

Exponent rules are fundamental guidelines that help us manage mathematical expressions involving powers. When dealing with exponents, one key rule to remember is the **Product of Powers rule**. This rule applies when multiplying numbers with the same base. The rule states that when we multiply two expressions with the same base, we simply add their exponents. For example, \(a^m imes a^n = a^{(m+n)}\). This means for numbers with the same base, you don't multiply the exponents; instead, you use addition. This simplification makes calculations much more straightforward and is always handy when dealing with powers in expressions.

Product of Powers

Now, let's explore the Product of Powers in more detail. Consider the expression like \(10^6 imes 10^{-3}\). Here, the base is 10, and we recognize that both terms involve powers of this same base. According to the Product of Powers rule, we add the exponents: \((10^6 imes 10^{-3}) = 10^{6 + (-3)} = 10^3\). It’s like combining two steps into one - maintain the base and work with the exponents to simplify the multiplication. This approach saves us time and effort in any expression involving exponents.

Multiplying Powers of 10

When multiplying powers of 10, the process is even more convenient because the base of 10 is particularly simple to work with. Powers of 10 are pervasive in scientific notation, as they allow for the easy expression of very large or very small numbers. For example, to multiply \(10^6\) and \(10^{-3}\), recognize these as powers with the same base of 10, and apply the Product of Powers rule: \(10^6 imes 10^{-3} = 10^3\). With the base 10 being standard, this makes the process of multiplying straightforward, facilitating the handling of both large and small numbers efficiently in scientific notation.

Base Numbers in Mathematics

Base numbers are the foundations of exponents. When we refer to the base, we are talking about the number being raised to a power. In mathematical expressions, the base is the number underneath the exponent, as seen in \(a^b\), where \(a\) is the base, and \(b\) is the exponent. Understanding base numbers is crucial because operations involving powers hinge on the base being the same for applying rules like the Product of Powers. In our problem, both 8 and 4 serve as our base numbers before applying powers of 10. Being able to identify and manipulate these bases in reference to their corresponding exponents is a powerful skill for solving complex exponential equations.