Problem 124 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\left(2.1 \times 10^{9}\right)\left(3.4 \times 10^{-4}\right)$$

Step-by-Step Solution

Answer

The simplified expression of \(2.1 \times 10^{9}\) and \(3.4 \times 10^{-4}\) is \(7.14 \times 10^{5}\)

Step 1: Identifying the Coefficients and Exponents

The numbers are given in scientific notation. In \(2.1 \times 10^{9}\), 2.1 is the coefficient and 9 is the exponent. And in \(3.4 \times 10^{-4}\), 3.4 is the coefficient and -4 is the exponent.

Step 2: Multiply the Coefficients

First, multiply the coefficients of the two numbers. Coefficient multiplication result is \(2.1 \times 3.4 = 7.14\)

Step 3: Adding the Exponents

Leaving behind the base \(10\), add the exponents of the two numbers. Exponent addition result here is \(9+(-4) = 5\)

Step 4: Writing the Final Answer

The result of multiplying the coefficients is the first part of the answer and the addition of exponents, attached to \(10\), is the second part of the answer. We write the answer in scientific notation. Combining both parts, we have: \(7.14 \times 10^{5}\).

Key Concepts

Multiplication of Exponents

When working with scientific notation, exponents tell us how many times the base number, typically 10, is used as a factor. Multiplying numbers in scientific notation involves dealing with these exponents. It's important to know the rule: when multiplying powers with the same base, you add the exponents together.

For example, consider the numbers in the exercise: \(10^{9}\) and \(10^{-4}\). Since they both have the base 10, we can apply the multiplication rule for exponents. Simply add the exponents \(9\) and \(-4\), resulting in an exponent of \(5\). So, \(10^{9} \times 10^{-4} = 10^{5}\).

This addition of exponents is crucial because it simplifies the process of multiplying large or tiny numbers by allowing us to focus on the exponents rather than the entire numbers themselves.

Coefficient Multiplication

The coefficients are the numbers in front of the base raised to a power in scientific notation. In the expression given in the exercise, we have two coefficients: 2.1 and 3.4. To simplify the expression, the first step is to multiply these coefficients together.

Multiplying coefficients is straightforward, just like multiplying regular numbers. In our example, we multiply 2.1 by 3.4, giving us a product of 7.14. This number will serve as the coefficient in our final simplified scientific notation form.

• Remember, the coefficients must be multiplied separately from the exponents.
• Always double-check multiplication to ensure accuracy in general problems.

Once we have our new coefficient, we are one step closer to expressing the answer in simplified scientific notation.

Exponent Addition

Adding exponents might seem a bit tricky at first, but it's actually a straightforward operation. Let's dive into why we add exponents when multiplying terms in scientific notation.

After obtaining the product of the coefficients, it's time to manage the exponential part of scientific notation. Both numbers in the exercise have a base of 10, so instead of multiplying the bases, we deal with the exponents by adding them. Here, we sum the exponents from \(10^{9}\) and \(10^{-4}\).

This operation uses the rule of multiplying exponents, which states that when multiplying like bases, you should add the exponents together. In this case, \(9 + (-4) = 5\). This new exponent of 5 is then coupled with our multiplied coefficient, forming the complete scientifically notated answer: \(7.14 \times 10^{5}\).

• Adding exponents is the key step that allows us to easily simplify multiplications in scientific notation.
• It takes advantage of the laws of exponents, making handling large numbers or very small ones manageable.