Problem 48 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(81x^{20}z^{8}y^{-32}\).

Step 1: Distribute the exponent

Distribute the exponent (4) to each term inside the parentheses: \[\left(\frac{3 x^{5} y^{-8}}{z^{-2}}\right)^{4} = \frac{(3^4) (x^5)^4 (y^{-8})^4}{(z^{-2})^4}\]

Step 2: Simplify the exponents

Simplify the exponents by applying the power rule, which states that \((a^m)^n = a^{mn}\): \[\frac{(3^4) (x^5)^4 (y^{-8})^4}{(z^{-2})^4} = \frac{3^{4} x^{5\cdot4} y^{-8\cdot4}}{z^{-2\cdot4}}\]

Step 3: Evaluate the exponents

Evaluate the exponents and simplify the expression: \[\frac{3^{4} x^{5\cdot4} y^{-8\cdot4}}{z^{-2\cdot4}} = \frac{81 x^{20} y^{-32}}{z^{-8}}\]

Step 4: Simplify negative exponents

Rewrite any terms with a negative exponent using the property \(a^{-n} = \frac{1}{a^n}\): \[\frac{81 x^{20} y^{-32}}{z^{-8}} = \frac{81 x^{20}}{y^{32}} \cdot z^{8}\]

Step 5: Combine the fractions and factors

Combine the fractions and factors to get the final simplified expression: \[\frac{81 x^{20}}{y^{32}} \cdot z^{8} = 81x^{20}z^{8}y^{-32}\] The simplified expression is: \[81x^{20}z^{8}y^{-32}\]

Key Concepts

Power Rule

The power rule is a fundamental principle of exponents. It simplifies expressions where one power is raised to another power. Imagine you have an expression like \((a^m)^n\). By using the power rule, you simplify it to \(a^{mn}\). This rule helps streamline computation, especially in algebraic expressions.

• Example: \((x^2)^3 = x^{2\cdot3} = x^6\). Here, the 2 and 3, the exponents, are multiplied, resulting in \(x^6\).
• In our exercise, \((x^5)^4\) translates into \(x^{5\cdot4} = x^{20}\).

Understanding this rule is vital for simplifying higher-level polynomial expressions. It reduces the complexity of dealing with multiple exponents. Remember to apply it whenever you encounter nested power terms.

Simplifying Expressions

Simplifying expressions is about rewriting them in the most condensed form. It involves using different algebraic rules. The goal is to make the expression cleaner and easier to work with.

• For instance, simplify \(3^4\) first. Compute \(3^4 = 81\).
• Then simplify the variable components: \(x^{20}\), \(y^{-32}\), and \(z^{-8}\).

In the example provided, the original expression involves multiple terms and powers. By breaking them down and applying rules like the power rule, the expression \( \left(\frac{3 x^{5} y^{-8}}{z^{-2}}\right)^4 \) becomes manageable. Eventually, the final outcome becomes straightforward as \(81x^{20}z^{8}y^{-32}\), showing all components clearly.

Negative Exponents

Negative exponents can be perplexing at first, but they follow a simple rule. When you encounter a negative exponent, convert it into a fraction to simplify. The rule is \(a^{-n} = \frac{1}{a^n}\). This property helps convert negative exponents into positive ones.

• Example: \(y^{-3}\) becomes \(\frac{1}{y^3}\).
• In our case, \(y^{-32}\) is rewritten as \(\frac{1}{y^{32}}\), and \(z^{-8}\) becomes \(\frac{1}{z^8}\).

By understanding and employing the rule of negative exponents, you gain control over the expression. In practice, this allows combining variables back into fractions to get a neat, simplified version. Mastering this will make handling complex expressions less daunting over time.