Problem 58 - Elementary Algebra

Question

\((3.14)\left(4 \times 10^{-1} \mathrm{~cm}\right)^{2}\left(2 \times 10^{-1} \mathrm{~cm}\right)\)

Step-by-Step Solution

Answer

0.10048 \text{cm}^3

Step 1: Simplify the base and exponent

First, simplify the term \((4 \times 10^{-1} \text{cm})^{2}\). This means squaring both the coefficient and the power of 10: \(4^2 \times (10^{-1})^2 = 16 \times 10^{-2} \text{cm}^2\).

Step 2: Multiply the constants

Next, multiply the constants together: \(3.14 \times 16 \times 2 \text{cm}^3\). This equals \(100.48 \text{cm}^3\).

Step 3: Multiply the powers of 10

Now, multiply the powers of 10: \(10^{-2} \times 10^{-1} = 10^{-3}\).

Step 4: Combine the results

Finally, combine the results from steps 2 and 3: \(100.48 \times 10^{-3} \text{cm}^3 = 0.10048 \text{cm}^3\).

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers concisely. It uses the format: \text{Coefficient} \times 10^{\text{Exponent}}. For example, in the problem, \(4 \times 10^{-1}\) has:

• A coefficient (\(4\))
• An exponent (\(-1\))

This notation simplifies calculations and makes it easier to read and compare numbers.
For instance, multiplying 4 by \(10^{-1}\) makes \(4\) smaller by a tenth, resulting in \(0.4\).
Using scientific notation can simplify our calculations, especially when dealing with powers.

Exponent Rules

Exponent rules are essential when working with scientific notation. Here are some key rules:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \(a^m \times a^n = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{mn}\)
• Zero Exponent Rule: \(a^0 = 1\)

In the problem, when squaring \(4 \times 10^{-1}\), we applied the power rule:
\begin{align*}(4 \times 10^{-1})^2 &= 4^2 \times (10^{-1})^2 \&= 16 \times 10^{-2}\text{cm}^2 \ \text{We then applied the product rule when multiplying:} \10^{-2} \times 10^{-1} \&= 10^{-3}\text{cm}^3 \ Understanding these rules helps simplify expressions and solve problems efficiently.

Multiplication

Multiplication with scientific notation involves multiplying the coefficients and then the powers of ten separately. Here's how it works in the problem:
1. First, simplify \(4 \times 10^{-1}\):

• Square it to get \(16 \times 10^{-2}\)

2. Multiply the constants:

• \(3.14 \times 16 \times 2 = 100.48\)

3. Multiply the exponents:

• \(10^{-2} \times 10^{-1} = 10^{-3}\)

Finally, we combined the results to get \(0.10048 \text{cm}^3\).
When dealing with multiplication, always handle coefficients and exponents separately for accuracy.