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

by THE IDEN 70 views

In mathematics, simplifying expressions is a fundamental skill. It allows us to present mathematical statements in their most concise and understandable form. This is particularly important when dealing with algebraic expressions that involve variables, coefficients, and exponents. In this article, we will walk through the step-by-step process of simplifying an expression that involves negative exponents and fractions, providing a clear and detailed explanation along the way.

Understanding the Problem

The expression we need to simplify is:

(βˆ’37mβˆ’4nβˆ’1)(βˆ’143m3n0)\left(-\frac{3}{7} m^{-4} n^{-1}\right)\left(-\frac{14}{3} m^3 n^0\right)

This expression involves several components:

  • Coefficients: The numerical values βˆ’37-\frac{3}{7} and βˆ’143-\frac{14}{3}.
  • Variables: The variables mm and nn.
  • Exponents: Both positive and negative exponents, such as βˆ’4-4, βˆ’1-1, 33, and 00.

To simplify this expression effectively, we need to understand and apply the rules of exponents and fractions. Let’s break down the key concepts before diving into the solution.

Key Concepts

Before we start simplifying, let's review the essential rules that govern exponents and fractions:

  1. Product of Powers Rule: When multiplying like bases, we add the exponents. Mathematically, this is expressed as amβ‹…an=am+na^m \cdot a^n = a^{m+n}.
  2. Negative Exponent Rule: A term raised to a negative exponent is equivalent to its reciprocal raised to the positive exponent. That is, aβˆ’n=1ana^{-n} = \frac{1}{a^n}.
  3. Zero Exponent Rule: Any nonzero number raised to the power of 0 is 1. Thus, a0=1a^0 = 1 if a≠0a \neq 0.
  4. Multiplying Fractions: To multiply fractions, we multiply the numerators and the denominators separately. That is, abβ‹…cd=aβ‹…cbβ‹…d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}.

With these rules in mind, we can systematically approach the simplification of the given expression.

Step-by-Step Simplification

Step 1: Multiply the Coefficients

The first step is to multiply the numerical coefficients together. We have:

(βˆ’37)β‹…(βˆ’143)\left(-\frac{3}{7}\right) \cdot \left(-\frac{14}{3}\right)

To multiply these fractions, we multiply the numerators and the denominators:

(βˆ’3)β‹…(βˆ’14)7β‹…3=4221\frac{(-3) \cdot (-14)}{7 \cdot 3} = \frac{42}{21}

Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 21:

4221=2\frac{42}{21} = 2

So, the product of the coefficients is 2.

Step 2: Multiply the Variables with Exponents

Next, we multiply the variables with their respective exponents. We have mβˆ’4β‹…m3m^{-4} \cdot m^3 and nβˆ’1β‹…n0n^{-1} \cdot n^0. We apply the product of powers rule, which states that amβ‹…an=am+na^m \cdot a^n = a^{m+n}.

For the variable mm:

mβˆ’4β‹…m3=mβˆ’4+3=mβˆ’1m^{-4} \cdot m^3 = m^{-4 + 3} = m^{-1}

For the variable nn:

nβˆ’1β‹…n0=nβˆ’1+0=nβˆ’1n^{-1} \cdot n^0 = n^{-1 + 0} = n^{-1}

Step 3: Combine the Results

Now, we combine the results from Step 1 and Step 2:

2β‹…mβˆ’1β‹…nβˆ’1=2mβˆ’1nβˆ’12 \cdot m^{-1} \cdot n^{-1} = 2m^{-1}n^{-1}

Step 4: Apply the Negative Exponent Rule

To eliminate the negative exponents, we use the rule aβˆ’n=1ana^{-n} = \frac{1}{a^n}.

mβˆ’1=1m1=1mm^{-1} = \frac{1}{m^1} = \frac{1}{m}

nβˆ’1=1n1=1nn^{-1} = \frac{1}{n^1} = \frac{1}{n}

So, we rewrite the expression as:

2β‹…1mβ‹…1n2 \cdot \frac{1}{m} \cdot \frac{1}{n}

Step 5: Final Simplification

Finally, we multiply the terms together:

2β‹…1mβ‹…1n=2mn2 \cdot \frac{1}{m} \cdot \frac{1}{n} = \frac{2}{mn}

Thus, the simplified expression is 2mn\frac{2}{mn}.

Final Answer

The simplified form of the expression (βˆ’37mβˆ’4nβˆ’1)(βˆ’143m3n0)\left(-\frac{3}{7} m^{-4} n^{-1}\right)\left(-\frac{14}{3} m^3 n^0\right) is 2mn\frac{2}{mn}.

Conclusion

Simplifying expressions with negative exponents involves a systematic approach using the rules of exponents and fractions. By breaking down the problem into smaller steps, we can handle complex expressions with ease. Remember to multiply coefficients, apply the product of powers rule, address negative exponents using the reciprocal, and combine the results to achieve the simplified form. This step-by-step methodology not only aids in solving the problem accurately but also enhances understanding of the underlying mathematical principles. Mastering these techniques is crucial for advanced mathematical studies and problem-solving in various fields.

Additional Practice

To further solidify your understanding, try simplifying the following expressions:

  1. (4xβˆ’2y3)(βˆ’2x5yβˆ’1)(4x^{-2}y^3)(-2x^5y^{-1})
  2. (12aβˆ’3b4)(6a2bβˆ’2)(\frac{1}{2}a^{-3}b^4)(6a^2b^{-2})
  3. (βˆ’5p4qβˆ’5)(3pβˆ’2q2)(-5p^4q^{-5})(3p^{-2}q^2)

By working through these examples, you will become more proficient in applying the rules of exponents and simplifying algebraic expressions.

Simplify the expression (βˆ’37mβˆ’4nβˆ’1)(βˆ’143m3n0)\left(-\frac{3}{7} m^{-4} n^{-1}\right)\left(-\frac{14}{3} m^3 n^0\right), assuming all variables are nonzero.

Simplifying Expressions with Negative Exponents A Step by Step Guide