Problem 69 - Algebra 2

Question

Which is the simplified form of \(\frac{4 x^{3} y^{2} z^{-1}}{\left(x^{-2} y^{3} z^{2}\right)^{2}} ?\) F. \(\frac{4 x^{7}}{y^{4} z^{5}}\) G. \(\frac{4 x y}{z^{5}}\) H. \(\frac{4}{y^{3} z^{5}}\) J. \(\frac{4}{x y^{4} z^{5}}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{4 x^{7}}{y^{4} z^{5}}\), option F.

Step 1: Simplify the Denominator by Distributing the Power

First, distribute the power of 2 that is outside the parenthesis to each variable term inside: \[ (x^{-2} y^{3} z^{2})^{2} = (x^{-2})^{2} (y^{3})^{2} (z^{2})^{2} \]. This simplifies to \[ x^{-4} y^{6} z^{4} \].

Step 2: Write the Entire Fraction

Combine the original numerator and the newly simplified denominator to form the overall fraction: \[ \frac{4 x^{3} y^{2} z^{-1}}{x^{-4} y^{6} z^{4}} \].

Step 3: Simplify the Variables in the Fraction

Use the exponent rules to simplify the fraction: - For \(x\): \[ x^{3} \div x^{-4} = x^{3 - (-4)} = x^{7} \]. - For \(y\): \[ y^{2} \div y^{6} = y^{2-6} = y^{-4} \]. - For \(z\): \[ z^{-1} \div z^{4} = z^{-1-4} = z^{-5} \].

Step 4: Combine and Simplify

Combine the results into a single fraction: \[ 4 \frac{x^{7}}{y^{4} z^{5}} \]. This can be expressed as: \[ \frac{4 x^{7}}{y^{4} z^{5}} \].

Key Concepts

Exponent Rules

Exponent rules are fundamental tools in algebra that help simplify expressions involving powers of numbers and variables. Let's look at a few essential exponent rules :

• Product of Powers: When multiplying like bases, you can add their exponents. For instance, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing like bases, you subtract the exponents. For example, \(a^m \div a^n = a^{m-n}\).
• Power of a Power: When you have an exponent raised to another exponent, you multiply the two exponents. For example, \( (a^m)^n = a^{m\times n} \).
• Negative Exponents: A negative exponent indicates a reciprocal. For instance, \(a^{-m} = \frac{1}{a^m}\).

In the given exercise, these rules help in simplifying the expressions significantly. By correctly applying the "quotient of powers" rule, you are able to efficiently reduce complex expressions to simpler forms. Recognizing how to handle powers can make solving equations less intimidating and more intuitive.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and/or the denominator. Understanding their simplification is crucial for issues involving algebraic expressions. Here are some pointers on working with them:

• Identify similar terms: Examining both the numerator and the denominator can reveal expressions that can cancel each other out or simplify easily.
• Factorize if necessary: Sometimes breaking down polynomials into a product of simpler factors can show clearly what can be canceled out, making simplification easy.
• Make use of exponent rules: As we saw, exponent rules like the quotient of powers can drastically change the form of the expression, simplifying it.

In the exercise, the original rational expression was simplified by distributing the power in the denominator and using the rules we've discussed. This shows how rational expressions, while they can look complex, boil down to simpler elements with the right approach.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. This process can make working with the expressions easier and provides clarity. Let's understand this better:

• Start by simplifying the components: Break down complex portions of expressions using rules of arithmetic and algebra, like exponent rules.
• Combine like terms: Identify and merge terms that have identical variables with the same exponents.
• Ensure everything is in its simplest form: Always check if the expression can be reduced further; sometimes rewriting parts exposes further opportunities for simplification.

For the exercise, simplification yielded the expression \( \frac{4 x^{7}}{y^{4} z^{5}} \). Instead of a complex, multilayered fraction, we end up with a sleek, manageable expression. Mastery of simplification in algebra isn't just about reducing complexity; it's about understanding your tools and using them effectively to foresee hidden equals and make computation efficient.