Problem 30 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\frac{7 m^{4}}{56 m^{2}}$$

Step-by-Step Solution

Answer

The short answer is: \[\frac{1}{8} m^{2}\]

Step 1: Divide the coefficients

Divide 7 by 56: \[\frac{7}{56} = \frac{1}{8}\] So, now we have: \[\frac{1}{8} \cdot \frac{m^{4}}{m^{2}}\]

Step 2: Simplify the variable's exponents

Subtract the exponent in the denominator from the exponent in the numerator: \(m^{4} ÷ m^{2} = m^{4-2} = m^{2}\)

Step 3: Combine the simplified parts

Now we have the simplified coefficient and the simplified variable part, so we combine them: \[\frac{1}{8} m^{2}\] The simplified expression is: \[\frac{1}{8} m^{2}\]

Key Concepts

Understanding Exponent Rules

Exponent rules are fundamental in simplifying algebraic expressions. When you divide powers with the same base, the rule is straightforward: subtract the exponents. If you have an expression like \(m^4 \div m^2\), you simply take the exponent from the numerator and subtract the one in the denominator: \(m^{4-2}\) which simplifies to \(m^2\). This rule helps simplify expressions efficiently and prevents errors when working with larger bases or more complex equations. Remember,

• Keep the base the same when subtracting exponents.
• Apply the formula \(a^m \div a^n = a^{m-n}\).

Always make sure the operations apply only to like terms, meaning terms that share the same base.

Dividing Coefficients

When simplifying fractions in algebraic expressions, dividing the coefficients is often your first step. Coefficients are the numerical part of terms, and simplifying them helps simplify the entire expression. In the original exercise, the coefficients were 7 and 56. Dividing these gives you a simpler fraction:

• Divide 7 by 56, resulting in \(\frac{1}{8}\).

This fraction represents the numerical part of your final simplified expression. It's crucial for helping you see the smaller, refined parts of a complex expression. Always make sure your coefficients are in their simplest form before moving on to handle the variable part of the expression.

Algebraic Fractions Simplification

Algebraic fractions can seem daunting at first, but they follow similar principles to numerical fractions. They involve both coefficients and variables. To simplify, perform operations like reducing coefficients and applying exponent rules to variables as demonstrated in the solution:

• First, simplify the coefficients as we did by dividing 7 by 56.
• Then, apply exponent rules to simplify the variables, reducing \(\frac{m^4}{m^2}\) to \(m^2\).

Once simplified, combine these parts to form the final expression. Practicing with algebraic fractions will improve your skills in both fractions and algebra. Always keep in mind:

• Perform operations separately on coefficients and variables.
• Combine your results to get the simplest form.