Problem 52 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\left(\frac{s^{2} t^{-4}}{5 s^{-1} t}\right)\) (b) \(\left(\frac{x y^{-2} z^{-3}}{x^{2} y^{3} z^{-4}}\right)^{-3}\)

Step-by-Step Solution

Answer

(a) \(\frac{s^3}{5 t^5}\); (b) \(\frac{x^3 y^{15}}{z^3}\)."

Step 1: Simplify the Expression (Part a)

First, look at the expression \( \frac{s^2 t^{-4}}{5 s^{-1} t} \). Simplify the powers of \(s\) and \(t\) using the rule \(a^m / a^n = a^{m-n}\).\\[s^2 / s^{-1} = s^{2 - (-1)} = s^{3}\] \[t^{-4} / t^{1} = t^{-4 - 1} = t^{-5}\]\ The expression becomes \( \frac{s^3 t^{-5}}{5}\).

Step 2: Eliminate Negative Exponents (Part a)

To remove the negative exponent from \(t^{-5}\), rewrite it as a positive exponent in the denominator: \[\frac{s^3}{5 t^5}\] This is the simplified expression for part (a).

Step 3: Simplify the Base Expression (Part b)

Consider \( \frac{x y^{-2} z^{-3}}{x^{2} y^{3} z^{-4}} \). Simplify each variable separately:1. \(\frac{x}{x^2} = x^{1-2} = x^{-1}\)2. \(\frac{y^{-2}}{y^{3}} = y^{-2-3} = y^{-5}\)3. \(\frac{z^{-3}}{z^{-4}} = z^{-3-(-4)} = z^{1}\)The expression simplifies to \(x^{-1} y^{-5} z^1\).

Step 4: Apply the Negative Exponent (Part b)

Now apply the \(-3\) exponent to the entire simplified expression \(x^{-1} y^{-5} z^{1}\):\[(x^{-1} y^{-5} z^{1})^{-3} = x^{-1\times (-3)} y^{-5\times (-3)} z^{1\times (-3)}\]Which simplifies to \(x^3 y^{15} z^{-3}\).

Step 5: Eliminate Negative Exponents (Part b)

Convert the negative exponent in \(z^{-3}\) into the denominator:\[x^3 y^{15} \frac{1}{z^3}\]This results in \(\frac{x^3 y^{15}}{z^3}\), which is the simplified expression for part (b).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in precalculus. It helps us transform complex terms into simpler, more manageable forms. In this exercise, we simplify expressions by combining like terms and reducing powers.

• Examine each part of the expression separately. Look at the numerators and denominators to see what you can simplify.
• Apply the rule: For any base \(a\), \(a^m / a^n = a^{m-n}\).
• Combine the coefficients (numbers) in front of variables like usual numbers.

For the expression \(\frac{s^2 t^{-4}}{5 s^{-1} t}\), simplify by combining the powers of \(s\) and \(t\) separately, following the power division rule above. This makes it easier to handle each variable distinctly.

Negative Exponents

Negative exponents can be intimidating, but they really just mean that you are dividing by that number or variable instead of multiplying. When you encounter a negative exponent, the rule is simple: move the base to the other side of the fraction line and change the exponent to positive.When you have \(t^{-5}\), it can be rewritten as \(\frac{1}{t^5}\). This step is crucial for simplifying expressions because it allows you to eliminate any negative exponents. Pay attention to the base and perform operations to convert negative exponents to a positive one.

• Any base with a negative exponent flips its position in the fraction. For example, \(x^{-1}\) becomes \(\frac{1}{x}\).
• Always apply these rules after simplifying other operations within the expression.

Algebraic Manipulation

Algebraic manipulation involves performing operations to restructure expressions in a simpler form. This includes distributing exponents over all terms, as shown in the negative exponent step of the exercise. For example, when applying the exponent \(-3\) to \((x^{-1} y^{-5} z^{1})\):

• Apply the exponent by multiplying it with each variable's power: \(x^{-1} \rightarrow x^{-1\times (-3)} = x^3\).
• Continue this method for all elements: \(y^{-5} \rightarrow y^{-5\times (-3)} = y^{15}\), and \(z^{1} \rightarrow z^{1\times (-3)} = z^{-3}\).

These steps help ensure that all bases are managed properly according to their signs. The key is consistently applying exponent rules and rechecking each transformation. By carefully manipulating these expressions, we achieve a more elegant and readable form. Just make sure to address all parts of the expression uniformly, check for further simplifications, and rewrite the final expression in its simplest, positive exponent form.