Problem 65 - Intermediate Algebra

Question

Perform each division. \(\frac{40 m^{17} n^{20}}{35 m^{15} n^{30}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{8m^2}{7n^{10}}\).

Step 1: Separate the Coefficients

First, focus on dividing the coefficients of the terms separately. The coefficients given are 40 and 35. Divide the coefficients: \[\frac{40}{35} = \frac{8}{7}\] This is because both numbers can be divided by their greatest common divisor, which is 5.

Step 2: Simplify the Variables with the Same Base

Next, simplify the variables by subtracting the exponents of like bases. Start with the variable \(m\).\[m^{17} \div m^{15} = m^{17-15} = m^{2}\]

Step 3: Simplify the Next Set of Variables

Now, simplify the variable \(n\).\[n^{20} \div n^{30} = n^{20-30} = n^{-10}\]A negative exponent indicates the reciprocal, so \(n^{-10} = \frac{1}{n^{10}}\).

Step 4: Write the Simplified Expression

Combine the results from the previous steps to write the simplified expression, including coefficients and simplified variables.\[\frac{8}{7} \times m^2 \times \frac{1}{n^{10}} = \frac{8m^2}{7n^{10}}\]

Step 5: Finalize the Division Expression

Ensure all parts of the expression are written as a single fraction:\[\frac{8m^2}{7n^{10}}\]This is the simplified result of the division.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number or variable. They tell us how many times we need to multiply a number by itself. When you see something like \(m^{17}\), it means \(m\) is being multiplied by itself 17 times.

• To simplify expressions with exponents, you frequently need to use the rule of exponent subtraction.
• This rule states that when you divide two numbers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.

For example, in \(m^{17} \div m^{15}\), the rule applies: you subtract 15 from 17, which equals 2. Thus, \(m^{17} \div m^{15} = m^2\).
Understanding this process is crucial for simplifying expressions and solving algebra problems effectively.

Algebraic Fractions and Their Roles

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. They are similar to numerical fractions but involve variables. When you encounter an exercise with algebraic fractions, it’s important to treat them with the same considerations you would with regular fractions. This includes seeking to simplify and understand each component of the fraction.

• For a fraction like \(\frac{40 m^{17} n^{20}}{35 m^{15} n^{30}}\), treat the numerator and denominator as separate expressions that can be simplified independently.
• Always check if the coefficients can be simplified by finding the greatest common divisor, which in this case is 5, simplifying the coefficient from \(\frac{40}{35}\) to \(\frac{8}{7}\).

After simplifying the coefficients, focus on simplifying the variables by applying the rules for exponents.
Once simplified, the algebraic fraction can reveal its simplest form, making it easier to understand and further manipulate if needed.

The Art of Simplification

Simplification in algebra is the process of making expressions more manageable or easier to comprehend. It often involves reducing fractions, combining like terms, and using rules of exponents.

• To simplify \(\frac{40 m^{17} n^{20}}{35 m^{15} n^{30}}\), the first step is to simplify the coefficients: \(\frac{40}{35}\) becomes \(\frac{8}{7}\).
• Next, handle the exponents by subtracting those in the denominator from those in the numerator for matching bases. For \(m\), we get \(m^2\), and for \(n\), a negative exponent \(n^{-10}\) transforms into \(\frac{1}{n^{10}}\).

Ultimately, putting all this together gives us the simplified fraction \(\frac{8m^2}{7n^{10}}\).
This example shows that simplification isn't merely about making things smaller, but about revealing the most efficient form.