Simplifying Expressions With Negative Exponents A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It's like decluttering a room – you organize and streamline to make things clearer and more manageable. Today, we'll tackle an expression that involves variables, coefficients, and those tricky negative exponents. Our mission is to simplify $6 x^{-8} u \\cdot 2 v^{-5} \\cdot 4 x^5 v^{-3} u^{-1}$, ensuring that our final answer boasts only positive exponents. Let's embark on this mathematical journey together.

Understanding the Basics

Before we dive into the heart of the problem, let's brush up on some essential concepts. Exponents are the shorthand notation for repeated multiplication. For instance, $x^3$ signifies $x \\cdot x \\cdot x$. A coefficient is the numerical factor that multiplies the variable part of a term (e.g., in $6x^{-8}u$, 6 is the coefficient). Variables, denoted by letters like $x$, $u$, and $v$, represent unknown quantities. The real game-changer here is the negative exponent. Remember, a negative exponent indicates the reciprocal of the base raised to the positive exponent. Mathematically, $x^{-n} = \\frac{1}{x^n}$. This rule is the key to eliminating negative exponents from our final expression.

The Power of Positive Exponents

Why are positive exponents so desirable? They provide a clear and intuitive representation of the quantity. A positive exponent tells us how many times to multiply the base by itself. Negative exponents, while mathematically valid, can be less straightforward to grasp at first glance. By converting them to positive exponents, we gain a clearer picture of the expression's value and behavior. In many contexts, particularly in scientific and engineering applications, expressions with positive exponents are the preferred form for communication and calculation. This is because they align with our fundamental understanding of multiplication and repeated operations.

Breaking Down the Expression

Now, let's dissect the expression $6 x^{-8} u \\cdot 2 v^{-5} \\cdot 4 x^5 v^{-3} u^{-1}$. We have three terms multiplied together. Each term consists of a coefficient and one or more variables raised to exponents, some of which are negative. Our strategy is to leverage the properties of exponents to combine like terms and transform negative exponents into positive ones. We will begin by rearranging the terms to group like variables together. This will make it easier to apply the product of powers rule, which states that when multiplying powers with the same base, you add the exponents.

The Strategy: A Step-by-Step Approach

Our roadmap for simplifying this expression involves several key steps. First, we'll rearrange the terms to group like variables together. This means bringing all the $x$ terms, $u$ terms, and $v$ terms into proximity. Second, we'll apply the product of powers rule ($x^m \\cdot x^n = x^{m+n}$) to combine the exponents of like variables. Third, we'll tackle the negative exponents by using the rule $x^{-n} = \\frac{1}{x^n}$ to move terms with negative exponents from the numerator to the denominator (or vice versa). Finally, we'll simplify the resulting expression, ensuring that all exponents are positive and the coefficients are multiplied together. This systematic approach will help us navigate the expression and arrive at the simplified form efficiently and accurately. Let's begin!

Step-by-Step Simplification

Step 1: Rearrange and Group Like Terms

The first step in simplifying our expression is to rearrange the terms so that like variables are grouped together. This makes it easier to combine them in the subsequent steps. We can use the commutative property of multiplication, which allows us to change the order of the factors without affecting the result. So, let's rewrite the expression:

$6 x^{-8} u \\cdot 2 v^{-5} \\cdot 4 x^5 v^{-3} u^{-1} = 6 \\cdot 2 \\cdot 4 \\cdot x^{-8} \\cdot x^5 \\cdot u \\cdot u^{-1} \\cdot v^{-5} \\cdot v^{-3}$

Now, we have clearly grouped the coefficients and the like variables together. This sets the stage for the next step, where we'll combine the coefficients and apply the product of powers rule to the variables.

Step 2: Combine Coefficients and Apply the Product of Powers Rule

Next, we combine the coefficients by multiplying them together: $6 \\cdot 2 \\cdot 4 = 48$. Now, let's focus on the variables. The product of powers rule states that when multiplying powers with the same base, you add the exponents: $x^m \\cdot x^n = x^{m+n}$. Applying this rule to our expression:

• $x^{-8} \\cdot x^5 = x^{-8+5} = x^{-3}$
• $u \\cdot u^{-1} = u^{1+(-1)} = u^0$
• $v^{-5} \\cdot v^{-3} = v^{-5+(-3)} = v^{-8}$

So, our expression now looks like this: $48 x^{-3} u^0 v^{-8}$.

Step 3: Handle Negative Exponents and Zero Exponents

We're on the home stretch! Now we need to address the negative exponents. Recall that $x^{-n} = \\frac{1}{x^n}$. This rule allows us to move terms with negative exponents from the numerator to the denominator (or vice versa) to make the exponents positive. Also, remember that any non-zero number raised to the power of 0 is 1 (i.e., $u^0 = 1$). Applying these rules:

• $x^{-3} = \\frac{1}{x^3}$
• $v^{-8} = \\frac{1}{v^8}$
• $u^0 = 1$

Substituting these back into our expression, we get: $48 \\cdot \\frac{1}{x^3} \\cdot 1 \\cdot \\frac{1}{v^8}$.

Step 4: Simplify to the Final Answer

Finally, we can combine the terms to arrive at our simplified expression: $48 \\cdot \\frac{1}{x^3} \\cdot 1 \\cdot \\frac{1}{v^8} = \\frac{48}{x^3 v^8}$.

Therefore, the simplified expression with only positive exponents is $\\frac{48}{x^3 v^8}$.

Conclusion: Mastering the Art of Simplification

Simplifying expressions, especially those with exponents, is a crucial skill in algebra and beyond. By understanding the rules of exponents and applying them systematically, we can transform complex expressions into more manageable forms. In this exercise, we successfully simplified $6 x^{-8} u \\cdot 2 v^{-5} \\cdot 4 x^5 v^{-3} u^{-1}$ to $\\frac{48}{x^3 v^8}$, a form that is both mathematically equivalent and easier to interpret. The key takeaways are the product of powers rule, the negative exponent rule, and the zero exponent rule. Practice applying these rules in various scenarios, and you'll become a master of simplification in no time!