Problem 35 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(m+7)^{4}(m-8)^{5}}{(m+7)^{7}(m-8)^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given rational expression: $$ \frac{(m+7)^{4}(m-8)^{5}}{(m+7)^{7}(m-8)^{2}} $$ Answer: $ (m+7)^{-3}(m-8)^{3} $

Step 1: Identify the exponential terms

In the given expression, we have two exponential terms: \((m+7)^{4}\) and \((m-8)^{5}\) in the numerator, and \((m+7)^{7}\) and \((m-8)^{2}\) in the denominator.

Step 2: Use the division properties of exponents

According to the properties of exponents, when dividing the same base raised to different exponents, we can subtract the exponent on the denominator from the exponent on the numerator. We will apply this rule to both \((m+7)\) and \((m-8)\) terms. For term \((m+7)\), the exponents are 4 in the numerator and 7 in the denominator. So, \((m+7)^{4} \div (m+7)^{7} = (m+7)^{4-7} = (m+7)^{-3}.\) For term \((m-8)\), the exponents are 5 in the numerator and 2 in the denominator. So, \((m-8)^{5} \div (m-8)^{2} = (m-8)^{5-2} = (m-8)^{3}.\)

Step 3: Simplify the expression

Now, we will substitute the simplified terms back into the expression: $$ \frac{(m+7)^{-3}(m-8)^{3}}{1} $$ The expression is now in its simplest form. So the simplified rational expression is: $$ \boxed{(m+7)^{-3}(m-8)^{3}} $$

Key Concepts

Simplifying Fractions

Imagine fractions as a relationship between two numbers. A fraction can often be simplified, or reduced, to make it easier to understand. Simplifying involves finding a simpler form that represents the same value. In algebra, a similar concept is used when dealing with rational expressions. Just as with numbers, the goal is to express the same value more simply. Let's look at how this works.

• Identify common factors: Check both the numerator and the denominator for common factors that can be divided out.
• Reduce step-by-step: For each common factor found, divide both the numerator and the denominator by this factor.
• Check your work: Ensure that no further simplification is possible and that the expression remains equivalent to the original.

In the example given, the common factors are related to the bases \(m + 7\) and \(m - 8\). By simplifying these using properties of exponents, we achieve the most reduced form of the expression.

Properties of Exponents

Exponents are powerful tools in mathematics, providing a concise way to express repeated multiplication. When simplifying expressions with exponents, understanding the properties of exponents can greatly aid the process. Let's explore some of these key properties.

• Quotient of Powers Property: When dividing like bases, subtract the exponents. For example, \(a^m \/ a^n = a^{m-n}\).
• Power of a Power Property: When raising a power to another power, multiply the exponents (\( (a^m)^n = a^{mn} \)).
• Zero Exponent Rule: Any non-zero base raised to the power of zero equals one (\( a^0 = 1 \)).
• Negative Exponent Rule: A negative exponent indicates a reciprocal (\( a^{-n} = \frac{1}{a^n} \)).

In the solution provided, the Quotient of Powers Property is used. We subtract the exponents of terms like \(m + 7\) and \(m - 8\) in both the numerator and denominator to simplify the expression.

Numerator and Denominator in Algebra

In any fraction, whether numerical or algebraic, there are two key parts: the numerator and the denominator. Understanding their roles is crucial in manipulating and simplifying expressions.

• Numerator: This is the top part of a fraction. It represents how many parts of the whole are being considered.
• Denominator: This is the lower part of a fraction. It represents the total number of equal parts the whole is divided into.
• Rational Expressions: In algebra, both the numerator and denominator are often polynomials. Simplifying involves working with these expressions, applying algebraic rules to find simpler forms.

Remember, when simplifying, any changes must be mirrored in both the numerator and the denominator to ensure the overall value remains unchanged. This balance is essential to maintaining the equality of the expression throughout the simplification process. That way, when you end up with an expression like \( (m+7)^{-3}(m-8)^3 \), you're sure it's truly simplified.