Problem 112 - College Algebra Essentials

Question

In Exercises \(111-114\), simplify each expression. Assume that all variables represent positive numbers. $$\left(8 x^{-6} y^{3}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6}$$

Step-by-Step Solution

Answer

The simplified expression is \(2x^3 / y\)

Step 1: Apply the power rule to each term inside the parentheses

The power rule states that to raise a power to a power, you should multiply the exponents. Therefore, \((8 x^{-6} y^{3})^{\frac{1}{3}}\) becomes \(2 x^{-2} y^{1}\) and \((x^{\frac{5}{6}} y^{-\frac{1}{3}})^{6}\) becomes \(x^5 y^{-2}\).

Step 2: Multiply the results from step 1.

To multiply terms with the same base, you should add the exponents. Therefore, \(2 x^{-2} y^{1}\) * \(x^5 y^{-2}\) simplifies to \(2 x^{3} y^{-1}\).

Step 3: Rewrite the solution in a form without negative exponents

Negative exponents indicate that the base is in the denominator of a fraction. Therefore, \(2 x^{3} y^{-1}\) becomes \(2x^3 / y\).

Key Concepts

Simplification in Algebra

Simplification is the process of reducing an algebraic expression to its simplest form. It's like tidying up a messy room. You want everything in the right place, as simple as possible.

• With simplification, we aim to make expressions easier to work with.
• The idea is to perform operations like multiplication, division, addition, and subtraction to combine like terms.

Simplifying expressions involves applying different algebraic rules. For instance, you use the distributive property, combine like terms, or apply rules of exponents. In our original exercise, simplification was crucial for making the expression neat and easy to interpret.
Practice is key. The more you work with simplification, the more natural it becomes. Don't be afraid to break down each step slowly when you're starting.

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. If you see \(x^2\), it means \(x\) multiplied by itself: \(x \times x\).

• The exponent tells us how many times to use the number in a multiplication.
• Exponents can be positive, negative, or even fractional.

In the original exercise, we see fractional exponents, like \( \left(8 x^{-6} y^3\right)^{\frac{1}{3}} \). A fractional exponent means root. \( x^{1/2} \) is the same as \( \sqrt{x} \).
Negative exponents are special too— they indicate division. For example, \( x^{-3} \) is equivalent to \( 1/x^3 \). Understanding these concepts helps us handle any exponent we might come across in algebraic expressions.

Algebraic Operations

Algebraic operations are the building blocks for solving expressions and equations. They involve fundamental manipulations like addition, subtraction, multiplication, and division of algebraic terms.
When dealing with operations involving exponents and variables:

• Multiply by adding their exponents if the bases are the same.
• Divide by subtracting their exponents if the bases are the same.

For example, in our exercise, we dealt with multiplying terms: \(2 x^{-2} y^1 \times x^5 y^{-2}\). The rule says we add exponents for bases that match:“\(x^{-2} \times x^5 \)” becomes \(x^{(-2 + 5)}\), which is \(x^3\).
Similarly, "\(y^1 \times y^{-2}\)" simplifies to \(y^{-1}\). Grasping these operations is crucial for mastering more complex algebra.