Problem 45 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\frac{6 y^{3} z}{2 y z^{2}}\) (b) \(\frac{\left(x y^{2} z^{3}\right)^{4}}{\left(x^{2} y^{2} z\right)^{3}}\)

Step-by-Step Solution

Answer

(a) \( \frac{3y^2}{z} \); (b) \( \frac{y^2 z^9}{x^2} \)."

Step 1: Simplify the Fraction for Part (a)

First, simplify the numerical coefficients in the fraction \( \frac{6 y^3 z}{2 y z^2} \). Divide 6 by 2, which equals 3. This gives us \( \frac{3 y^3 z}{y z^2} \).

Step 2: Simplify the Variables for Part (a)

Apply the quotient rule for exponents to simplify where possible. For \( y \), subtract the exponents: \( y^{3-1} = y^2 \). For \( z \), subtract the exponents: \( z^{1-2} = z^{-1} \). This results in \( 3y^2z^{-1} \).

Step 3: Eliminate Negative Exponents for Part (a)

Rewrite the expression to eliminate negative exponents. The term \( z^{-1} \) becomes \( \frac{1}{z} \). Therefore, \( 3y^2z^{-1} = \frac{3y^2}{z} \).

Step 4: Simplify Powers for Part (b)

For part (b), apply the power rule to the numerator and denominator separately. Begin with the numerator: \( (x y^2 z^3)^4 = x^4 y^8 z^{12} \). Next, simplify the denominator: \( (x^2 y^2 z)^3 = x^6 y^6 z^3 \).

Step 5: Simplify the Fraction for Part (b)

Combine the two expressions into a single fraction: \( \frac{x^4 y^8 z^{12}}{x^6 y^6 z^3} \). Apply the quotient rule for each variable. For \( x \), the exponents are \( 4 - 6 \), resulting in \( x^{-2} \), for \( y \), \( 8 - 6 \) gives \( y^2 \), and for \( z \), \( 12 - 3 \) gives \( z^9 \).

Step 6: Eliminate Negative Exponents for Part (b)

To eliminate the negative exponent for \( x \), rewrite \( x^{-2} \) as \( \frac{1}{x^2} \). The expression becomes \( \frac{y^2 z^9}{x^2} \).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in mathematics. It involves reducing expressions to their simplest form without changing their value. Simplification makes equations easier to handle and more understandable. Let's look at the crucial steps involved:

• **Combine Like Terms**: Look for terms that have the same variables raised to the same power and combine them by adding or subtracting their coefficients.
• **Apply Mathematical Operations**: Perform operations such as addition, subtraction, multiplication, and division on the coefficients and variables.
• **Use Exponent Rules**: Use the rules of exponents, such as multiplying and dividing powers, to simplify further.

Simplifying algebraic expressions can involve various steps depending on the complexity of the expression. It helps in solving equations more efficiently and quickly.

Negative Exponents

Understanding negative exponents is crucial for simplifying expressions effectively. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In other words:

• For any non-zero number or variable \( a \) and a positive integer \( n \), \( a^{-n} = \frac{1}{a^n} \).
• Negative exponents can appear in both the numerator and the denominator of a fraction. Moving a negative exponent across the fraction bar changes the sign of the exponent.

For example, in step 3 of the solution, the expression \( z^{-1} \) was transformed into \( \frac{1}{z} \). This is crucial for making an expression free of negative exponents, which aids in achieving its simplest form.Negative exponents, therefore, help express a number in a more standard form, which is particularly useful in equations and fractions.

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental exponent rule used when dividing two expressions with the same base. The rule states that when you divide two powers with the same base, you simply subtract the exponent of the denominator from the exponent of the numerator:

• For any base \( a \), and exponents \( m \) and \( n \), the rule is given by \( \frac{a^m}{a^n} = a^{m-n} \).
• This rule simplifies divisions involving powers and is crucial for simplifying complex fractions.

For example, in step 2 and step 5 of the solution, the quotient rule was used to simplify expressions like \( y^{3-1} = y^2 \) and \( x^{4-6} = x^{-2} \). The application of this rule is pivotal in reducing complicated algebraic expressions into more manageable forms by lowering the power of the variable wherever possible. Understanding and applying the quotient rule accurately is a powerful tool for solving algebraic problems efficiently.