Problem 35 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols when necessary. See Example 2 . $$ \left(-27 n^{9}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-3n^3\).

Step 1: Understand the Expression

We are given the expression \((-27 n^9)^{1/3}\). The goal is to simplify this expression. It involves both a negative number inside the parenthesis and a variable raised to a power.

Step 2: Apply the Power of a Power Rule

When an expression inside a parenthesis is raised to a power, each component inside the parenthesis is also raised to that power. So, \((-27 n^9)^{1/3} = (-27)^{1/3} (n^9)^{1/3}\).

Step 3: Simplify the Cube Root of -27

The cube root of a negative number is also negative, and the cube root of 27 is 3. Thus, \((-27)^{1/3} = -3\).

Step 4: Simplify the Variables with Power

Apply the power to the variable: \((n^9)^{1/3}\). Use the rule \((a^m)^{n} = a^{m \, n}\). Therefore, \(n^{9 \cdot \frac{1}{3}} = n^{3}\).

Step 5: Combine the Results

Combine the simplified results from Step 3 and Step 4. Thus, \((-27 n^9)^{1/3} = -3n^3\).

Key Concepts

Understanding Cubed Roots

Cubed roots are a fascinating part of the mathematics universe! To understand the cubed root, you need to know that it is the opposite operation of cubing a number. When you cube a number, you multiply it by itself three times. Taking the cubed root moves in the other direction. For example, the cube root of 27 is a number that, when multiplied by itself three times, equals 27. In this case, the number is 3 because \(3 \times 3 \times 3 = 27\). Now, what happens if you need a cube root of a negative number like -27?

• The cube root of a negative number remains negative. So, \((-27)^{1/3} = -3\).
• This is because \(-3 \times -3 = 9\ and \ 9 \times -3 = -27\).

This property of having a negative cubed root only applies to cubed roots (and not even roots like square roots), which makes them unique and interesting.

The Power of a Power Rule

The "power of a power" rule is one of those essential tools that makes simplifying expressions much easier. It tells us how to handle expressions where an exponent is raised to another exponent.The rule states that when you have a base raised to a power, and that whole term is raised to another power, you simply multiply the exponents. This is written as \((a^m)^n = a^{m \cdot n}\). In simpler terms:

• You keep the base the same.
• You multiply the exponents together.

Using this rule with a variable such as \((n^9)^{1/3}\), you multiply 9 by \(\frac{1}{3}\). This results in the exponent 3, simplifying the expression to \(n^3\).This rule is very handy, especially when simplifying complex expressions with multiple layers of exponents. It makes calculations quicker and easier.

Working with Variable Exponents

Variable exponents might seem intimidating initially, but once you get the hang of them, they're straightforward! They show how many times the base variable is multiplied by itself.In the expression, you often need to perform operations that involve these exponents, such as raising to another power or taking roots. In our exercise:

• We saw \((n^9)^{1/3}\). Applying the power of a power rule transformed it into \(n^3\).
• This simplifies the expression beautifully from a more complex form to a simple and elegant result.

Understanding how to work with variable exponents helps in managing algebraic expressions and equations effectively, whether they involve arithmetic, geometry, or more advanced fields. And they are everywhere in algebra!