Problem 77 - Algebra 2 and Trigonometry

Question

In \(74-82,\) write each expression as a power with positive exponents in simplest form. $$ \left(\frac{8 a^{2} z^{6}}{27 x^{9} a^{-4} z^{-1}}\right)^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2 a^{2} z^{\frac{7}{3}}}{3 x^{3}} \).

Step 1: Simplify Inside the Parentheses

First, we'll simplify the expression inside the parentheses. We have \( \frac{8 a^{2} z^{6}}{27 x^{9} a^{-4} z^{-1}} \). Simplify the powers of \( a \) and \( z \) by subtracting exponents: \( a^{2 - (-4)} = a^{6} \) and \( z^{6 - (-1)} = z^{7} \). The expression now becomes \( \frac{8 a^{6} z^{7}}{27 x^{9}} \).

Step 2: Distribute the Cube Root

Next, apply the cube root (or raise to the \( \frac{1}{3} \) power) to each component of the fraction separately. This gives \[ (\frac{8 a^{6} z^{7}}{27 x^{9}})^{\frac{1}{3}} = \frac{(8)^{\frac{1}{3}} (a^{6})^{\frac{1}{3}} (z^{7})^{\frac{1}{3}}}{(27)^{\frac{1}{3}} (x^{9})^{\frac{1}{3}}}. \]

Step 3: Simplify Each Term

Evaluate each term separately: - \( (8)^{\frac{1}{3}} = 2 \) because \( 8 = 2^3 \).- \( (a^{6})^{\frac{1}{3}} = a^{6 \times \frac{1}{3}} = a^{2} \).- \( (z^{7})^{\frac{1}{3}} = z^{7 \times \frac{1}{3}} = z^{\frac{7}{3}} \).- \( (27)^{\frac{1}{3}} = 3 \) because \( 27 = 3^3 \).- \( (x^{9})^{\frac{1}{3}} = x^{9 \times \frac{1}{3}} = x^{3} \).

Step 4: Combine Simplified Terms

Now combine the simplified terms into a single fraction: \( \frac{2 a^{2} z^{\frac{7}{3}}}{3 x^{3}} \). This expression is now in its simplest form with positive exponents.

Key Concepts

Fractional Exponents

The term fractional exponents might seem daunting at first, but they are quite straightforward once you break them down. These exponents are just a way to represent roots. For instance, if you see the expression \( a^{\frac{1}{3}} \), it's the same as the cube root of \( a \).
A fractional exponent follows the principle that \( a^{\frac{m}{n}} \) is equal to \( \sqrt[n]{a^m} \). Here's how it works in simpler terms:

• The denominator of the fraction, \( n \), indicates the root we are taking: square root, cube root, etc.
• The numerator, \( m \), tells us how many times to raise the base \( a \) to a power after taking the root.

Understanding fractional exponents is crucial as it allows you to simplify expressions that contain root operations, helping streamline complex algebraic tasks into simpler forms. In our exercise, applying a fractional exponent simplifies calculating cube roots, such as converting \( (8)^{\frac{1}{3}} \) to \( 2 \). This knowledge makes complex expressions manageable, focusing only on the essentials.

Simplifying Expressions

Simplifying expressions is all about making them as clean and simple as possible. It means reducing the complexity wherever you can. Let's break down how simplification works, especially with fractions and exponents:
Start by examining each part of the expression.

• Look at numbers and coefficients, and reduce them if possible. In the exercise, \( \frac{8}{27} \) remains as is since neither are divisible by the same number.
• For variables with the same base appearing in both numerator and denominator, subtract their exponents: \( a^{2} / a^{-4} = a^{6} \) since subtracting a negative is like adding.
• Group terms together before applying any operations. It's easier to handle smaller parts that make up the whole.

Finally, express the cleaned-up expression using positive exponents. Always try to bring the variables to the numerator or denominator where deemed necessary, turning negative exponents into positive by moving terms across the fraction. Simplifying creates clarity, reducing room for error or confusion when working with algebraic expressions.

Positive Exponents

Working with positive exponents generally simplifies mathematical expressions and makes them easier to interpret and solve. Positive exponents tell you how many times to multiply a base by itself. In contrasts with negative exponents, which are just tools to indicate division.
To convert negative exponents into positive ones, simply use the rule of moving the base across the fraction bar. For instance, \( a^{-4} \) becomes \( 1/a^4 \). This technique is vital to simplifying expressions and is commonly used in algebra.

• When an entire expression is raised to an exponent, apply it to each term individually.
• Fractions can have exponents applied separately to the numerator and denominator, as our resolved example shows.

Through these approaches, you maintain positive exponents, avoiding the confusion negative signs might bring. Positive exponents make it clear how multiplication setups work in algebraic expressions, presenting a straightforward picture without involving complex inversions or potentially leading to calculation errors.