Problem 92 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\left(125 x^{9} y^{6}\right)^{1 / 3}$$

Step-by-Step Solution

Answer

The simplified expression of \(\left(125 x^{9} y^{6}\right)^{1 / 3}\) using properties of exponents is \(5x^{3}y^{2}\).

Step 1: Apply Cubic Root to Each Term Inside the Parenthesis

The first step is to simplify the expression inside the parentheses, treating each term separately. We can apply the cubic (third) root to each, which gives us \(\sqrt[3]{125}x^{9/3}y^{6/3}\).

Step 2: Simplify Each Term

Once the cubic root has been applied to each term, we can further simplify. The cubic root of 125 is 5. Dividing the exponents of x and y by 3 also simplifies those terms. Therefore, we have \(5x^{3}y^{2}\).

Step 3: Final Answer

Our final answer after applying the rules of exponents and simplifying is \(5x^{3}y^{2}\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest possible form while maintaining their original value. The main goal is to make the expression easier to understand or work with.

When simplifying expressions, especially those with multiple terms or operations, look for opportunities to:

• Combine like terms
• Apply arithmetic operations
• Use algebraic identities or properties

Simplifying expressions with properties of exponents can further reduce complexity. For example, if you have terms like \(x^a \cdot x^b\), you can simplify it to \(x^{a+b}\). Similarly, when dividing terms like \(x^a \/ x^b\), you subtract the exponents: \(x^{a-b}\).

In the exercise, the expression \(\left(125 x^{9} y^{6}\right)^{1 / 3}\) was simplified by applying the cubic root to each term and using the properties of exponents. This simplification results in a much more manageable form, \5x^{3}y^{2}\.

Cubic Root

The cubic root of a number or term is a value that, when multiplied by itself three times, equals the original number or term. It's the opposite of cubing a number.

For example, the cubic root of 8 is 2, since \(2 \times 2 \times 2 = 8\). Similarly, the cubic root of 125 is 5, because \(5 \times 5 \times 5 = 125\).

When simplifying expressions using a cubic root, it's often useful to break down coefficients and constants to make the calculation easier. You can apply the cubic root to each part of the expression independently, as seen with \(\sqrt[3]{125}x^{9/3}y^{6/3}\).

This results in taking the cubic root of each base and simplifying the exponents separately. This process can seem daunting at first, but practice will help it become more intuitive!

Exponent Rules

Exponent rules are the mathematical guidelines that help us simplify expressions involving powers of numbers. These rules make handling complex algebraic expressions much easier.

Some of the most commonly used exponent rules include:

• The product rule: \(a^m \cdot a^n = a^{m+n}\)
• The quotient rule: \(a^m \/ a^n = a^{m-n}\)
• The power of a power rule: \( (a^m)^n = a^{m \cdot n} \)

Using these rules helps break down the expressions into simpler parts. They allow expressions like \(x^{9/3}\) to be rewritten as \(x^3\), by dividing the exponent by 3.

In the provided exercise, exponent rules were essential to simplify the terms once the cubic root was applied. Each exponent inside the parentheses was divided by 3, leading to the final simplified expression: \5x^{3}y^{2}\. With consistent application of these rules, even complex expressions become manageable.