Problem 20 - Elementary Algebra

Question

Perform each multiplication. $$ \left(2.1 \times 10^{-9}\right)\left(3 \times 10^{-11}\right) $$

Step-by-Step Solution

Answer

Answer: The result of the multiplication is \(6.3 \times 10^{-20}\).

Step 1: Multiply the coefficients

First, multiply the coefficients of the numbers, which are 2.1 and 3: $$ (2.1)(3) = 6.3 $$

Step 2: Multiply the powers of 10

Next, multiply the powers of 10, which are \(10^{-9}\) and \(10^{-11}\). When multiplying powers of 10 with the same base, add the exponents: $$ 10^{-9} \times 10^{-11} = 10^{-9 + (-11)} = 10^{-20} $$

Step 3: Combine the results

To find the final answer, combine the results from Steps 1 and 2: $$ (6.3)(10^{-20}) $$ So the final result of the given multiplication is: $$ \left(2.1 \times 10^{-9}\right)\left(3 \times 10^{-11}\right) = 6.3 \times 10^{-20} $$

Key Concepts

Multiplying Coefficients

In scientific notation, the numeral aspect of the expression, often referred to as the coefficient, can be treated like any other number in multiplication. For example, when we have two numbers in scientific notation, say \(2.1 \times 10^{-9}\) and \(3 \times 10^{-11}\), we first address the coefficients 2.1 and 3. Multiply these as normal numbers:

\[2.1 \times 3 = 6.3\]
Remember, the focus here is on the coefficients only, disregarding the powers of ten for the moment. This step is crucial for understanding how each part of a scientific notation is handled separately in the multiplication process.

Exponent Rules

Regarding exponent rules, an essential concept in scientific notation is managing the powers of ten. When we multiply two powers of ten, the exponents are added together, as long as the base, which is 10, is the same. For instance, when multiplying \(10^{-9}\) and \(10^{-11}\), we apply the exponent rule:

\[10^{-9} \times 10^{-11} = 10^{-9 + (-11)} = 10^{-20}\]
This rule simplifies calculations significantly and applies to any base, not just ten. This handling of exponents reflects the fundamental principle that multiplying like bases results in the exponents being summed.

Powers of Ten

The powers of ten are a cornerstone of scientific notation. They help express very large or very small numbers compactly. In scientific notation, a number is written as a product of a coefficient and a power of ten, emphasizing the role of the coefficient in representing the significant figures of the number, while the power of ten shows the scale or magnitude. This format is especially useful in fields like science and engineering, where extreme values are commonly dealt with. For example, \(10^2\) signifies the number 100, while \(10^{-2}\) represents 0.01. This system allows the straightforward comparison of magnitudes and simplifies calculations involving numbers with large discrepancies in size.

Algebraic Problem Solving

Algebraic problem solving encompasses a strategic approach to tackling mathematical challenges. When working with scientific notation, it's beneficial to break down the problem into manageable steps. As seen in the exercise, performing separate operations for coefficients and exponents helps simplify the process. Upon obtaining the results for each part, they're combined to achieve the final answer. This methodical practice not only yields a correct solution but also enhances comprehension of the underlying mathematical principles, allowing students to apply similar tactics to a variety of problems.