Problem 48 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) \( \frac{x^{10}}{y z^4} \); (b) \( r^9 s^2 \)."}

Step 1: Expand the Exponents

Begin by expanding the expression \( \left(\frac{2 x^{3} y^{2}}{z^{3}}\right)^{2} \). Apply the power of a quotient and power of a power rules: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \) and \( (a^m)^n = a^{m \times n} \). This expansion gives us: \( \frac{(2)^2 (x^3)^2 (y^2)^2}{(z^3)^2} = \frac{4 x^6 y^4}{z^6} \).

Step 2: Multiply the Expressions

Now, multiply the simplified expression from Step 1 with the first term \( \frac{x^4 z^2}{4 y^5} \). Use the rule of multiplying fractions: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \). This results in \( \frac{x^4 z^2}{4 y^5} \times \frac{4 x^6 y^4}{z^6} = \frac{4x^{10}y^4z^2}{4y^5z^6} \).

Step 3: Simplify the Fractions

Cancel common factors in the numerator and denominator. Here, the 4 in the numerator and denominator can be canceled out, leaving: \( \frac{x^{10}y^4z^2}{y^5z^6} \).

Step 4: Simplify the Exponents

Simplify the expression by applying the quotient rule for exponents \( \frac{a^m}{a^n} = a^{m-n} \). For each variable:- \( x: x^{10}\) remains since there is no \( x \) in the denominator.- \( y: \frac{y^4}{y^5} = y^{4-5} = y^{-1} \)- \( z: \frac{z^2}{z^6} = z^{2-6} = z^{-4} \)This yields: \( x^{10}y^{-1}z^{-4} \).

Step 5: Remove Negative Exponents

To eliminate negative exponents, rewrite the terms with negative exponents in the denominator as positive exponents: \( \frac{x^{10}}{y^1z^4} \).

Step 6: Simplify Part (b): Expand and Simplify

Simplify the expression \( \frac{(r s^2)^3}{(r^{-3} s^2)^2} \). First, expand: - Numerator: \((r s^2)^3 = r^3 s^6 \)- Denominator: \((r^{-3} s^2)^2 = r^{-6} s^4 \)Thus, the expression becomes \( \frac{r^3 s^6}{r^{-6} s^4} \).

Step 7: Simplify Part (b): Simplify Exponents and Remove Negative Exponents

Simplify using the quotient rule for exponents:- \( r: \frac{r^3}{r^{-6}} = r^{3-(-6)} = r^{9} \)- \( s: \frac{s^6}{s^4} = s^{6-4} = s^2 \)Resulting in: \( r^9 s^2 \).

Key Concepts

Exponentiation

Exponentiation is a foundational concept in precalculus that involves raising a number or expression to a certain power. When dealing with powers, it's important to become familiar with several key rules:

• Product of Powers Rule: When multiplying two expressions with the same base, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Power of a Quotient Rule: Apply the exponent to both the numerator and the denominator: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).

These rules help transform complicated expressions into simpler forms, especially when dealing with fractions or multiple terms within an expression. In the given exercise, these exponentiation rules are used to expand and simplify parts of the initial expression, making it more manageable.

Simplifying Expressions

Simplifying expressions is a crucial step in precalculus, making equations easier to interpret and solve. Simplification involves reducing an expression to its simplest form, often by combining like terms, factoring, or canceling. In our exercise, several steps were involved:

• Expand Exponents: Before mixing terms, always expand exponents fully based on the rules of exponentiation.
• Combine Like Terms: Multiply expressions properly, ensuring all terms are combined using their respective rules.
• Cancel Common Factors: Look for common factors in numerators and denominators, and cancel them out to declutter the expression.

These strategies streamline the expression, reducing complexity and paving the way for further manipulations or conclusions, such as solving equations or evaluating function values. In the exercise, simplifying fractions and combining powers without negative signs is essential to reach the final form.

Negative Exponents

Negative exponents can seem tricky, but they simply represent the reciprocal of the positive exponent. In other words, rather than multiply by a base, you divide by it:

• Negative Exponents Rule: \( a^{-n} = \frac{1}{a^n} \).
• Simplify to Eliminate: Convert negative exponents by moving the term to the opposite part of the fraction (from numerator to denominator, or vice versa) to make the exponent positive.

Dealing with negative exponents in an expression, as shown in the given exercise step, involves rewriting terms with positive exponents for clearer interpretation. This approach helps eliminate negatives, simplifying calculations and preventing computational mistakes. By understanding and applying this concept, you're equipped to handle complex expressions by reducing them to simpler, more manageable forms with only positive exponents.