Problem 28 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(2 x^{2} y^{-5}\right)\left(6 x^{-3} y\right)\left(\frac{1}{3} x^{-1} y^{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{x^{2}y}\).

Step 1: Apply the Product Rule for Exponents

The product rule for exponents states that when you multiply like bases, you add the exponents. We will apply this rule separately for the variable components: \(x\) and \(y\).Start with the \(x\) terms: \[(2x^2y^{-5})(6x^{-3}y)(\frac{1}{3}x^{-1}y^3)\]Take the product of exponents for \(x\):\[x^{2} \cdot x^{-3} \cdot x^{-1} = x^{2 + (-3) + (-1)} = x^{-2}\]

Step 2: Simplify the Exponents for y

Apply the product rule for the \(y\) terms: \[y^{-5} \cdot y^{1} \cdot y^{3} = y^{-5 + 1 + 3} = y^{-1}\]

Step 3: Simplify the Coefficients

Multiply the coefficients separately from the variables:Start with multiplying the first two coefficients:\[2 \cdot 6 = 12\] Then multiply by the third coefficient:\[12 \cdot \frac{1}{3} = 4\]

Step 4: Combine All Simplified Expressions

Combine the simplified coefficients and exponents:\[4x^{-2}y^{-1}\]This is a simplified expression of the given product in terms of positive exponents where applicable. However, if negative exponents are present, they can be written as fractions.

Step 5: Express Negative Exponents as Fractions

Convert negative exponents into fractions:The exponents \(x^{-2}\) and \(y^{-1}\) can be expressed as fractions:\[\frac{4}{x^{2}y}\] This is the fully simplified form of the given expression.

Key Concepts

Product Rule for Exponents

When working with exponents, understanding the product rule is essential. This rule states that when you multiply numbers or expressions that have the same base, you can simply add the exponents together. For example, if you have \(a^m \times a^n\), the result is \(a^{m+n}\).
In our exercise, we applied the product rule to the variable \(x\) and \(y\) terms separately.

• For \(x\): \(x^2 \times x^{-3} \times x^{-1} = x^{2 + (-3) + (-1)} = x^{-2}\).
• For \(y\): \(y^{-5} \times y^1 \times y^3 = y^{-5 + 1 + 3} = y^{-1}\).

The key takeaway here is to always check that the bases match before applying this rule. By keeping the base the same and operating only on the exponents, calculations become much simpler and less error-prone.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and performing arithmetic operations to make the expression as concise as possible. In algebra, simplifying is crucial for solving equations effectively.
In the given exercise, after applying the product rule for exponents, we proceeded to simplify further by handling coefficients separately from the variables.

• Coefficients \(2\), \(6\), and \(\frac{1}{3}\) were multiplied together to get \(4\).
• The expression then simplified to \(4x^{-2}y^{-1}\).

Remember to multiply coefficients in sequence, similar to how you apply the product rule to exponents. This step-by-step approach ensures no part of the expression is overcomplicated or left unsimplified. Keeping work organized and labeling parts of the equation can help avoid mistakes.

Negative Exponents

Negative exponents can be a little tricky at first, but they are easy to handle once you grasp their meaning. A negative exponent indicates that the base should be taken as the reciprocal. For example, \(a^{-n} = \frac{1}{a^n}\). This transformation can help simplify expressions, especially when final results are expected in certain formats.
In our expression, the terms included negative exponents \(x^{-2}\) and \(y^{-1}\). These were expressed as:

• \(x^{-2}\) becomes \(\frac{1}{x^2}\).
• Similarly, \(y^{-1}\) transforms to \(\frac{1}{y}\).

Hence, our fully simplified result was \(\frac{4}{x^2y}\). Always remember that moving from a negative exponent to its reciprocal format can make complex expressions easier to read and utilize in further mathematical operations.