Problem 46 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\frac{2 x^{3} y^{4}}{x^{5} y^{3}}\) (b) \(\frac{\left(2 v^{3} w\right)^{2}}{v^{3} w^{2}}\)

Step-by-Step Solution

Answer

(a) \( \frac{2y}{x^2} \); (b) \( 4v^3 \).

Step 1: Identify the Expression

Let's examine the given expression (a) \( \frac{2x^3 y^4}{x^5 y^3} \). We need to simplify this expression by eliminating any negative exponents.

Step 2: Simplify the Coefficients and Variables Separately

Start with the coefficients. Here, the coefficient is '2', and we don't divide it by anything, so it remains '2'. Next, simplify the variable \( x \): \( x^{3-5} = x^{-2} \), and \( y \): \( y^{4-3} = y^1 \).

Step 3: Eliminate Negative Exponents

For \( x^{-2} \), move \( x^2 \) to the denominator to turn the exponent positive. The expression becomes \( \frac{2y}{x^2} \).

Step 4: Finalize Simplified Expression for Part (a)

The simplified expression for part (a) with positive exponents is \( \frac{2y}{x^2} \).

Step 5: Identify the Second Expression

Now analyze the expression (b) \( \frac{(2v^3w)^2}{v^3w^2} \). Begin by expanding the power of the numerator.

Step 6: Expand and Simplify the Numerator in Expression (b)

Expand \( (2v^3w)^2 = 2^2(v^3)^2(w)^2 \). So, the expansion is \( 4v^6w^2 \).

Step 7: Simplify the Expanded Expression

Now simplify by dividing \( 4v^6w^2 \) by \( v^3w^2 \). For \( v \), calculate \( v^{6-3} = v^3 \). For \( w \), calculate \( w^{2-2} = \) (no \( w \) term). The result is \( 4v^3 \).

Step 8: Finalize Simplified Expression for Part (b)

The simplified expression for part (b) is \( 4v^3 \).

Key Concepts

Understanding Negative Exponents

Negative exponents can initially seem confusing, but they are actually quite straightforward once you get the hang of them. When we have a negative exponent like \( x^{-a} \), it simply means you take the reciprocal of the base with a positive exponent. In simpler terms, \( x^{-a} \) is the same as \( \frac{1}{x^a} \). This means that any negative exponent can be converted into a positive by placing the base in the denominator.
Here’s a quick breakdown:

• \( x^{-1} \) becomes \( \frac{1}{x} \)
• \( x^{-2} \) becomes \( \frac{1}{x^2} \)
• General rule: \( x^{-a} = \frac{1}{x^a} \)

In our example, \( x^{-2} \) was transformed into \( \frac{1}{x^2} \) to get rid of the negative exponent, simplifying the expression as much as possible.

Decoding Variable Exponents

Variable exponents can be a bit tricky, as they involve both the variable and the power to which it is raised. When simplifying, we often deal with dividing or multiplying terms with the same base, which requires some specific rules. If you remember these rules, it will be much easier to handle such expressions with confidence.
Handling variable exponents:

• When multiplying, add the exponents: \( x^a \times x^b = x^{a+b} \).
• When dividing, subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).
• If a variable is raised to a negative power, change it to a positive by flipping it to the denominator or numerator, depending on its original position.

In step (a) of our problem, we used these rules: subtracting exponents for \( x \) and \( y \) during division, leading to \( x^{-2} \) and \( y^1 \), which were later simplified to ensure positive exponents only.

Mastering Mathematical Expression Simplification

Simplifying mathematical expressions involves making them as concise as possible without changing their value. This is crucial in solving complex problems more easily and is a fundamental skill in math. Here's how you can approach simplification:
Steps for simplifying:

• Identify similar terms: terms that have the same base and power. Simplifying them might lead to a more concise expression.
• Apply the rules for exponents: use addition or subtraction of exponents as we previously discussed to simplify terms.
• Focus on eliminating negative exponents to convert them into positive for easier reading and understanding.
• Always perform any numeric simplification, such as squaring a coefficient or combining like terms.

In the original solution: For part (b), we expanded \((2v^3w)^2\) as \(4v^6w^2\), before simplifying it to \(4v^3\) by using these techniques. Instead of keeping the negative or redundant terms, we made sure every part of the variable and number was simplified to its simplest form by following these principles. With practice, simplifying expressions becomes second nature and highly rewarding.