Problem 87 - College Algebra

Question

In Exercises 87–106, perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientifi c notation answer to two decimal places. $$\left(3 \times 10^{4}\right)\left(2.1 \times 10^{3}\right)$$

Step-by-Step Solution

Answer

The final solution to the multiplication problem is \(6.3 \times 10^{7}\)

Step 1: Multiplication of Base Numbers and Addition of Exponents

To solve this problem, we first need to multiply the base numbers, 3 and 2.1. We also add the exponent parts together. The new base is \(3 \times 2.1 = 6.3\) and the new exponent is \(4 + 3 = 7\).

Step 2: Write the Final Result in Scientific Notation

Our final result is then \(6.3 \times 10^{7}\). We don't need to round up to two decimal places in this case because the base already comes as a number with only one decimal place.

Key Concepts

Multiplication

Multiplication is one of the fundamental arithmetic operations, crucial when dealing with numbers, especially in scientific notation. In the original exercise, multiplication is used to combine two base numbers as part of converting expressions to scientific notation. To multiply numbers, you simply find the product of two or more numbers, often referred to as the 'factors.'

• For example, in \((3 \times 10^{4}) (2.1 \times 10^{3})\), you multiply the base numbers first: \(3 \times 2.1\), which equals \(6.3\).
• This product becomes the new base in the scientific notation format.
• Multiplying the base numbers is a straightforward process that results in a single value, simplifying complex scientific calculations.

Remember, multiplication is just repeated addition. When you multiply, you're adding a number (called the "multiplicand") to itself a certain number of times (equal to the "multiplier"). By practicing with different exercises, you can quickly improve your ability to handle multiplications, making scientific computations become much clearer.

Exponents

Exponents are shorthand for repeated multiplication and play a vital role in the world of mathematics. They are used in scientific notation to indicate how many times a base number is multiplied by itself. This is extremely helpful in representing both very large and very small numbers concisely. In the exercise, the expression \(10^{4} \) and \(10^{3} \) are examples of exponents. Exponents show us that the base (in this case, 10) is multiplied by itself the number of times indicated by the exponent value.

• For instance, \(10^{4} = 10 \times 10 \times 10 \times 10 = 10,000\).
• When multiplying two numbers in scientific notation, you must add the exponents: \(4 + 3 = 7\).
• This means you have \(10^{7} \), which is \(10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10,000,000\).

Understanding how to work with exponents is vital for performing complex mathematical operations efficiently. It simplifies the representation and comparison of large-scale calculations.

Base Numbers

Base numbers refer to the fundamental numerical part of an expression in scientific notation. They form the base for the power of 10 when numbers are expressed in this compact format. This is important because it helps in easily writing and calculating large and small numbers in a concise manner. In scientific notation, the base is typically a decimal number greater than or equal to 1 and less than 10.

• In the exercise case, the base numbers are 3 and 2.1.
• Multiplying these gives \(6.3\), which is the new base number in the final scientific notation.
• The equation becomes \(6.3 \times 10^{7}\).

Base numbers are key because they retain the significant figures of the original number while simplifying computations by maintaining a consistent framework that revolves around powers of 10. Through this understanding, tasks involving very large or small numbers become manageable and understandable.