Problem 37 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$5^{2} m \div 5^{0} m$$

Step-by-Step Solution

Answer

Simplified form is \(5^2 m\).

Step 1: Understand the Problem

We are given the expression \(5^{2} m \div 5^{0} m\) and need to simplify it, expressing the final answer with positive exponents.

Step 2: Apply Division Property of Exponents

The division property of exponents states \(\frac{a^m}{a^n} = a^{m-n}\). Here, we apply it to the terms containing 5. We have \(5^2 \div 5^0 = 5^{2-0} = 5^2\).

Step 3: Simplify the Coefficient

Since \(5^2\) is already the simplest form with positive exponents, and \(m\) is unaffected by the division since it appears identically in numerator and denominator, our expression simplifies to \(5^{2}\cdot m\).

Step 4: Final Simplification

Multiply the remaining terms to see that the expression remains \(\mathbf{5^2 m}\), which is already expressed with positive exponents and cannot be simplified further.

Key Concepts

Division Property of Exponents

In algebra, the division property of exponents is a useful tool when simplifying expressions with the same base. It helps you to divide expressions efficiently without having to perform long multiplication or division. The division property states that dividing terms with the same base can be simplified by subtracting their exponents:

• For example, if you have \(a^m \div a^n\), you can simplify this as \(a^{m-n}\).

In this exercise, the expression \(5^{2} m \div 5^{0} m\) requires using this property:

• Both parts have the base '5'.
• You subtract the exponent of the denominator (0) from the exponent in the numerator (2).

This gives us \(5^{2-0} = 5^{2}\). It’s essential to remember to only subtract the exponents of terms with the same base. This method avoids unnecessary complications and gives your simplified result directly.
Understanding this property is key to mastering exponent operations and makes it easier to handle more complex algebraic expressions.

Positive Exponents

Positive exponents indicate the number of times a base is multiplied by itself. In contrast, negative exponents indicate the reciprocal. When simplifying expressions, it is generally best practice to express them with positive exponents, as this form is typically considered the simplest and most standard.

• Positive exponents make interpretation and further calculations easier.
• They are less prone to mistakes compared to negative exponents which require taking the reciprocal.

For instance, in the original exercise, after simplifying using division property of exponents, we ended up with \(5^2\). \(5^2\) is a straightforward positive exponent expression indicating \(5 \times 5 = 25\).Writing in positive exponents often aligns with modern algebraic conventions, making your work more aligned with standard practices. This makes verifying your work easier in educational settings or when using further in calculations.

Coefficient Simplification

Simplifying coefficients in algebra involves reducing terms to their simplest form without altering their value. Coefficients are the numerical part of terms in an expression. In the provided exercise, the problem involved simplifying the expression \(\"5^2 m \div 5^0 m\"\).

• The coefficient \(5^2\) needs simplification but is already simple enough (li>It’s crucial to recognize when further simplification isn’t necessary, as \(m\) stays the same.
• Keep the variables intact where possible unless they cancel each other.

In equations involving coefficients and variables, simplifying can often also involve evaluating numerical parts (like \(5^2\), which equals 25), but in this case, it's left in exponential form for clarity and relevance to additional operations. Recognizing these nuances in simplification makes solving algebraic problems more straightforward and keeps work efficiently clear and accurate.