Problem 56 - Elementary Algebra

Question

\((3.14)\left(1.8 \times 10^{-5} \mathrm{~km}\right)^{2}\)

Step-by-Step Solution

Answer

1.01736 \(\times\) 10^{-9} \mathrm{km}^2

Step 1: Identify the Variables

Identify the variables in the expression: the constant \(3.14\) and the term \(1.8 \times 10^{-5} \mathrm{km}\).

Step 2: Square the Term

Square the term \(1.8 \times 10^{-5} \mathrm{km}\). Remember to square both the number and the exponent: \[ (1.8 \times 10^{-5})^2 = (1.8)^2 \times (10^{-5})^2 = 3.24 \times 10^{-10} \mathrm{km}^2 \]

Step 3: Multiply by the Constant

Multiply the squared term by the constant \(3.14\): \[ 3.14 \times 3.24 \times 10^{-10} \mathrm{km}^2 \]

Step 4: Calculate the Final Value

Perform the multiplication: \[ 3.14 \times 3.24 = 10.1736 \] and then adjust for the exponent: \[ 10.1736 \times 10^{-10} = 1.01736 \times 10^{-9} \mathrm{km}^2 \]

Key Concepts

Exponents

Understanding exponents is crucial for solving problems like the given exercise. An exponent tells us how many times to multiply a number by itself. For example, in the term \(10^{-5}\), '-5' is the exponent, and it means we divide 1 by 10, five times. When numbers with exponents are squared, you multiply both the base and the exponent. For instance, squaring \(1.8 \times 10^{-5}\) results in \( (1.8)^2 \times (10^{-5})^2 = 3.24 \times 10^{-10} \). Next, we use this knowledge to simplify expressions easily.

Multiplication

Multiplication is another foundational concept in this exercise. Here, we multiply constants and numbers with exponents separately. For instance, after squaring \(1.8 \), we get \(3.24 \) and then multiply it by 3.14. This step-by-step approach ensures accuracy. Next, combine the product with your exponent. For example, \(3.14 \times 3.24 = 10.1736\), and then combine with the exponent part, resulting in \(10.1736 \times 10^{-10} \).

Powers of Ten

Powers of ten simplify large or tiny numbers, often seen in scientific notation. For example, \(10^5 \) equals 100,000, while \(10^{-5} \) equals 0.00001. This concept is essential for expressing the final answer in a manageable form. In our example, \(10.1736 \times 10^{-10} \) is transformed into \(1.01736 \times 10^{-9} \) by adjusting the decimal point. This makes interpretation and further calculations easier.