Problem 88 - Elementary Algebra

Question

\(f=\frac{\left(1.34 \times 10^{1}\right)^{2}}{\left(5 \times 10^{1}\right)(9.8)}\)

Step-by-Step Solution

Answer

0.366

Step 1: Simplify the Numerator

First, simplify the numerator \( \ (1.34 \times 10^{1})^{2} \). This means you raise 1.34 to the power of 2 and multiply the exponent by 2: \( \ (1.34^2) \times 10^{(1 \times 2)} = 1.7956 \times 10^{2}. \)

Step 2: Simplify the Denominator

Next, simplify the denominator \( \ (5 \times 10^{1}) \times 9.8 \). First, multiply the constants: \( \ 5 \times 9.8 = 49. \) Then include the exponent: \( 49 \times 10^{1}. \)

Step 3: Combine and Simplify the Fraction

Now combine the simplified numerator and denominator: \( \ \frac{1.7956 \times 10^{2}}{49 \times 10^{1}}. \) Separate the constants and the powers of 10: \( \ \frac{1.7956}{49} \times \frac{10^{2}}{10^{1}}. \)

Step 4: Final Calculation

Calculate the two fractions separately. First, \( \ \frac{1.7956}{49} \approx 0.0366. \) Then, \( \frac{10^{2}}{10^{1}} = 10. \) Finally, combine these results: \( 0.0366 \times 10 = 0.366. \)

Key Concepts

Numerator and Denominator Simplification

Simplifying fractions often requires breaking down both numerators and denominators. Let's start with the numerator in this example. We have: ewline ewline \[(1.34 \times 10^1)^2\]. The first step is to address the power, which means raising 1.34 to the power of 2: ewline ewline \[(1.34)^2 = 1.7956.\] Then, we take care of the exponent, multiplying \(10^1 \) by 2: ewline ewline \[10^{1 \times 2} = 10^2.\] Combining these, the simplified numerator is: ewline ewline \[1.7956 \times 10^2.\] ewline Now let’s move on to the denominator: ewline ewline \[(5 \times 10^1) \times 9.8.\] First, multiply the constants: ewline ewline \[5 \times 9.8 = 49.\] Then, include the exponent: ewline ewline \[49 \times 10^1.\] Finally, our fraction becomes: ewline ewline \[\frac{1.7956 \times 10^2}{49 \times 10^1}.\] ewline This step is crucial because simplifying both parts makes the entire fraction easier to work with.

Powers and Exponents

Understanding powers and exponents is fundamental in dealing with scientific notation and other mathematical operations. In this exercise: ewline ewline \[(1.34 \times 10^1)^2\] shows how to handle both base numbers and exponents. First, you square 1.34: ewline ewline \[1.34^2 = 1.7956.\] Now, manage the exponent separately by noting: ewline ewline \[10^{1 \times 2} = 10^2.\] This results in \[1.7956 \times 10^2.\] ewline ewline Dealing with the denominator’s powers, we have \[(5 \times 10^1) \times 9.8\]. Multiply the constants first getting: ewline ewline \[5 \times 9.8 = 49,\] then remembering the exponent: ewline ewline \[49 \times 10^1.\] Combining these two separate steps efficiently helps simplify the equation: ewline ewline \[\frac{1.7956 \times 10^2}{49 \times 10^1}.\] Keeping the powers and exponents separate until the final steps can make the calculations easier and reduce mistakes.

Fraction Operations

When dealing with fractions, the process of simplifying can ease the calculations. In the given problem, we end up with: ewline ewline \[\frac{1.7956 \times 10^2}{49 \times 10^1}.\] Separately handle the constants and the exponents: ewline ewline \[\frac{1.7956}{49} \times \frac{10^2}{10^1}.\] First, divide the constants: ewline ewline \[\frac{1.7956}{49} ≈ 0.0366.\] Next, divide the powers of 10 remembering the law of exponents: ewline ewline \[\frac{10^2}{10^1} = 10^{2-1} = 10.\] Finally, multiply the constant result with the power of 10: ewline ewline \[0.0366 \times 10 = 0.366.\] Understanding these steps simplifies working with complex fractions and ensures accurate calculations in exercises involving scientific notation and exponentiation.