Problem 85 - Elementary Algebra

Question

\(M=\frac{(4)(3.14)\left(1 \times 10^{-7}\right)(6)}{2(3.14)\left(5 \times 10^{-3}\right)}\)

Step-by-Step Solution

Answer

M = 2.4 \times 10^{-4}

Step 1 - Identify the formula

The given formula to solve is: \[M = \frac{(4)(3.14)(1 \times 10^{-7})(6)}{2(3.14)(5 \times 10^{-3})}\]

Step 2 - Simplify constants

Simplify the constants. Notice that both the numerator and denominator have a common factor, 3.14. Divide both by 3.14: \[M = \frac{(4)(1 \times 10^{-7})(6)}{2(5 \times 10^{-3})}\]

Step 3 - Multiply the numerators

Multiply the constants in the numerator: \[(4)(1 \times 10^{-7})(6) = 24 \times 10^{-7}\]

Step 4 - Multiply the denominators

Multiply the constants in the denominator: \[2(5 \times 10^{-3}) = 10 \times 10^{-3}\]

Step 5 - Divide the values

Divide the numerator by the denominator: \[M = \frac{24 \times 10^{-7}}{10 \times 10^{-3}} = \frac{24}{10} \times \frac{10^{-7}}{10^{-3}}\]

Step 6 - Simplify the powers of 10

Simplify the division of powers of 10: \[M = 2.4 \times 10^{-7 + 3} = 2.4 \times 10^{-4}\]

Key Concepts

Simplifying Expressions

Simplifying expressions means making them easier to understand or solve. In our example, we do this by canceling out common factors and combining like terms. One of the first steps was to notice that both the numerator and the denominator had the factor 3.14. By dividing both by 3.14, we made the expression simpler. This is a common way to simplify algebraic expressions:

• Look for common factors in the numerator and denominator.
• Cancel them out to reduce the complexity of the expression.
• Combine like terms to further simplify.

Each of these steps makes the problem easier to solve and helps you avoid mistakes.

Powers of 10

Powers of 10 are a shorthand way to express very large or very small numbers. In our example, we have numbers like \(1 \times 10^{-7}\) and \(5 \times 10^{-3}\). These numbers use powers of 10 to show how many times to multiply 10 by itself. For instance, \(10^{-7}\) means \(1 \/ 10^7\). When simplifying, remember these key points:

• \(10^a \times 10^b = 10^{a+b}\).
• \(10^a \/ 10^b = 10^{a-b}\).
• Using these rules, you can simplify expressions involving powers of 10.

This makes your calculations much easier and helps you quickly see the relationship between the numbers.

Algebraic Fractions

Algebraic fractions are fractions where both the numerator and the denominator are algebraic expressions. In our example, the fraction looks complex, but we can make it simpler by following some straightforward steps:

• Identify common factors in the numerator and the denominator.
• Cancel these common factors to simplify each part.
• Perform any necessary multiplications or divisions to combine terms.

For example, simplifying the numerator \(4(1 \times 10^{-7})(6) = 24 \times 10^{-7}\) and the denominator \(2(5 \times 10^{-3}) = 10 \times 10^{-3}\) helps reduce the fraction to a simpler form.

Numerator and Denominator

In any fraction, the numerator is the top part, and the denominator is the bottom part. Understanding these parts is crucial when simplifying algebraic fractions:

• The numerator tells you how many parts you are considering.
• The denominator tells you how many equal parts the whole is divided into.

In our exercise, the original fraction was \( \frac{(4)(3.14)(1 \times 10^{-7})(6)}{2(3.14)(5 \times 10^{-3})} \). By simplifying each part, we made it easier to solve:

• Simplify common factors like 3.14.
• Multiply the constants in both the numerator and the denominator.
• Simplify powers of 10 to get the final result.

This step-by-step approach helps make the problem more manageable.