Problem 21 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(8 x^{4} y^{-3}\right)\left(\frac{1}{2} x^{-5} y^{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{xy}\).

Step 1: Group Like Terms

Start by grouping the terms with the same variables. Consider the expression: \( \left(8 x^{4} y^{-3}\right)\left(\frac{1}{2} x^{-5} y^{2}\right) \). Break it down into: \((8 \cdot \frac{1}{2}) (x^{4} \cdot x^{-5}) (y^{-3} \cdot y^{2})\) for easier simplification.

Step 2: Simplify the Coefficients

Simplify the coefficients. Multiply \(8\) and \(\frac{1}{2}\) to get \(4\). So now we have: \(4 (x^{4} \cdot x^{-5}) (y^{-3} \cdot y^{2})\).

Step 3: Apply the Exponent Rules for \(x\)

Use the rule \(x^a \cdot x^b = x^{a+b}\). For \(x\): \(x^{4} \cdot x^{-5} = x^{4 + (-5)} = x^{-1}\).

Step 4: Apply the Exponent Rules for \(y\)

Similarly, apply the exponent rule to \(y\). For \(y\): \(y^{-3} \cdot y^{2} = y^{-3+2} = y^{-1}\).

Step 5: Write the Final Expression

Combine the simplified terms to write the final expression. We have \(4 \cdot x^{-1} \cdot y^{-1}\), which can also be written as \(\frac{4}{x \cdot y}\), since negative exponents denote the reciprocal.

Key Concepts

Exponent Rules

When dealing with exponents, it is essential to understand the rules that govern their behavior. These rules simplify expressions and help us manipulate terms with exponents effortlessly.

Here are some key exponent rules:

• Multiplying with the same base: If you multiply terms with the same base, you add the exponents. For instance, if you have \(x^a \cdot x^b\), the result is \(x^{a+b}\).
• Dividing with the same base: When dividing, subtract the exponents, so \(\frac{x^a}{x^b} = x^{a-b}\).
• Power of a power: Raising a power to another power means multiplying the exponents: \((x^a)^b = x^{a \cdot b}\).
• Zero exponent: Any base raised to the power of zero is equal to one, \(x^0 = 1\).

Understanding these rules will make simplifying expressions like the original exercise much easier!

Negative Exponents

Negative exponents might seem confusing at first, but they just indicate that the base is on the other side of a fraction. In other words, a negative exponent tells you to take the reciprocal of the base.

Here's what you need to know:

• Reciprocal meaning: \(x^{-a} = \frac{1}{x^a}\).
• Negative to positive: To make negative exponents positive, move the base to the opposite part of the fraction (numerator to denominator or vice versa).

When encountering negative exponents in an expression such as the one from the exercise, you can rewrite them to better understand and simplify the term.

Product of Powers

The product of powers rule is a convenient and powerful tool in algebra that helps you efficiently simplify and combine terms with exponents.

In your original exercise, you encounter terms like \(x^4 \cdot x^{-5}\), where the product of powers rule comes into play:

• When multiplying, simply add the exponents together. In the example: \(x^4 \cdot x^{-5} = x^{4 + (-5)} = x^{-1}\).
• This rule also applies to other variables like \(y\) in the exercise: \(y^{-3} \cdot y^2 = y^{-1}\).

Using the product of powers rule allows for clearer and more accurate simplification of expressions with multiple like bases.

Reciprocal of Exponents

The reciprocal of exponents is a vital concept that stems from understanding negative exponents. When a term contains a negative exponent, you can express it as the reciprocal of the positive exponent.

For instance, from your exercise:

• \(x^{-1} = \frac{1}{x}\)
• \(y^{-1} = \frac{1}{y}\)

This technique of conversion helps in transforming complex expressions into simpler, more manageable forms. By turning negative exponents into reciprocals, you can see the expression in a different light, ultimately rewriting it neatly, as in the exercise solution: \(\frac{4}{x \cdot y}\).

Mastering this concept makes handling expressions with multiple negative exponents much easier.