Problem 46 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{12}{25 s^{5}} \div \frac{10}{15 s^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{18}{25 s^3} \).

Step 1: Write the Division as Multiplication

The first step in dividing fractions is to rewrite the division problem as a multiplication problem. For \( \frac{12}{25 s^{5}} \div \frac {10}{15 s^{2}} \), we write it as: \[ \frac{12}{25 s^{5}} \times \frac {15 s^{2}}{10} \].

Step 2: Multiply the Fractions

Now, multiply the numerators together and the denominators together:Numerator: \(12 \times 15 s^2 = 180 s^2\)Denominator: \(25 s^5 \times 10 = 250 s^5\)So, we have: \[ \frac{180 s^2}{250 s^5} \].

Step 3: Simplify the Numerical Part

Simplify the numerical components of the fraction. Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 10:\( \frac{180}{10} = 18 \)\( \frac{250}{10} = 25 \)Thus, the fraction becomes:\[ \frac{18 s^2}{25 s^5} \].

Step 4: Simplify the Variable Part

Now simplify the variable part. Divide the powers of \(s\) in the numerator and denominator using the law of exponents \(a^{m}/a^{n} = a^{m-n}\):\( s^{2-5} = s^{-3} \)Thus, the expression simplifies to:\[ \frac{18}{25 s^3} \].

Key Concepts

Multiplying Fractions

When dealing with fraction division, it's crucial to recognize that dividing by a fraction is the same as multiplying by its reciprocal. This is one of the fundamental properties of fractions. So, instead of thinking about division, we think about multiplication with a flipped fraction.

For instance, in the problem \[\frac{12}{25s^5} \div \frac{10}{15s^2}\], we convert division to multiplication: \[\frac{12}{25s^5} \times \frac{15s^2}{10}\].

The idea is simple:

• Flip the second fraction (the divisor).
• Replace the division sign with a multiplication sign.

This approach makes it easier to process mathematically, as multiplication is often more straightforward than division with fractions. Once you have the multiplication equation, you can multiply the numerators and denominators to get a new fraction. This brings us to the next step in simplification.

Simplifying Expressions

Simplification is about making an expression look neater or simpler. After multiplying fractions, you often end up with a complex expression. Your goal is to break it down into its simplest form.

Let's consider the result from our multiplication: \[\frac{180s^2}{250s^5}\]. Here, simplification involves two parts:

• Numerical Simplification: Identify the greatest common divisor (GCD) of the numbers in the numerator and denominator. For 180 and 250, the GCD is 10. Divide both by this number:
  Numerator: \(\frac{180}{10} = 18\)
  Denominator: \(\frac{250}{10} = 25\)
• Variable Simplification: Use the law of exponents to handle the variable part (the \(s\) terms). Here, you apply the rule \(a^{m}/a^{n} = a^{m-n}\).

Simplifying means both reducing the numbers and correctly managing the variable terms to achieve the simplest expression possible.

Law of Exponents

The law of exponents is a powerful tool for simplifying expressions involving powers. When you divide expressions with the same base, you subtract the exponents. For instance, if you have \(s^2\) in the numerator and \(s^5\) in the denominator, calculate the exponent of \(s\) in the resulting expression by subtracting:

\[s^{2-5} = s^{-3}\]

This result means:

• You subtract 5 (the exponent in the denominator) from 2 (the exponent in the numerator).
• A negative exponent like \(s^{-3}\) implies that \(s^3\) is actually in the denominator.

Thus, it transforms our expression to
\[\frac{18}{25s^3}\].
Understanding this basic law helps in accurately manipulating expressions and reducing them effectively. Always remember: subtraction of exponents handles division within the same base, simplifying complex fractional expressions neatly.