Problem 15 - Algebra 1: Concepts and Skills

Question

Use the power of a quotient property to simplify the expression. $$ \left(\frac{-5}{m}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{25}{m^{2}}\).

Step 1: Application of Power of Quotient Rule

According to the rule of power of a quotient, when an entire fraction is raised to a power, we raise the numerator and the denominator to that power separately. Here, as per the rule, each part, i.e., -5 and m inside the brackets are squared separately, which leads to \(\left(-5\right)^{2}\) and \(m^{2}\) respectively.

Step 2: Squaring Each Part

Now, we square each part. Squaring -5 gives us 25, and squaring m gives us \(m^{2}\). Hence, \(\left(-5\right)^{2} = 25\) and \(m^{2} = m^{2}\).

Step 3: Rewriting the Simplified Expression

The simplified form of the given expression can be rewritten as \(\frac{25}{m^{2}}\).

Key Concepts

Simplifying Expressions

Understanding how to simplify expressions is crucial in algebra. The process involves reducing an algebraic expression to its simplest form, making it easier to interpret and work with. For instance, when simplifying the expression \(\left(\frac{-5}{m}\right)^{2}\), one must deal with both the numerator and the denominator. Simplifying goes both ways—not just turning complex expressions into simple ones, but also understanding the steps for how an expression was made simple. Here, it's not just about applying the exponent; it's about recognizing that both components of the fraction, the -5 and the m, are subjected to the squaring process independently. This individual attention to each component is what ultimately simplifies the expression to \(\frac{25}{m^{2}}\).

Exponents and Powers

Exponents and powers represent a basic, yet powerful concept used extensively in algebra. They refer to the number of times a base is multiplied by itself. In the expression \(\left(\frac{-5}{m}\right)^{2}\), '2' is the exponent, which indicates that the fraction \(\frac{-5}{m}\) needs to be multiplied by itself once, as any number to the power of 2 squares the number.

Working with Negative Bases

It's essential to recognize that when squaring a negative number, the result is positive because multiplying two negative numbers together gives a positive. Hence, \(\left(-5\right)^{2} = 25\). Similarly, when squaring a positive base, as in the denominator m, the positive characteristic of the base is retained: \(m^{2}\).

Algebraic Rules

Algebraic rules are the backbone of manipulating mathematical expressions. For example, the power of a quotient property is an algebraic rule that comes into play when a fraction is raised to an exponent. This property states that the exponent applies to both the numerator and the denominator independently.

Following this rule ensures that expressions are simplified correctly, maintaining the mathematical integrity of the original problem. An understanding of these rules, including the power of a quotient, helps students solve expressions accurately and paves the way for tackling more complex algebraic equations.