Problem 110 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \left(-27 x^{3}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x\).

Step 1: Recognize the Cube Root

The expression \((-27 x^{3})^{1/3}\) involves taking the cube root of the entire expression inside the parentheses.

Step 2: Apply the Power Rule

When simplifying expressions like \((a \, b)^{n}\), apply the power to each factor inside separately: \((a^n \, b^n)\). Therefore, \((-27 x^3)^{1/3}\) becomes \((-27)^{1/3} \cdot (x^3)^{1/3} \).

Step 3: Simplify \((-27)^{1/3}\)

The cube root of \(-27\) is \(-3\) because \((-3)^3 = -27\). Hence, \((-27)^{1/3} = -3\).

Step 4: Simplify \((x^3)^{1/3}\)

The expression \((x^3)^{1/3}\) can be simplified using the property \((a^m)^n = a^{m \cdot n}\). This gives \(x^{3 \cdot \frac{1}{3}} = x^{1} = x\).

Step 5: Combine the Simplified Parts

Now combine the results from the previous steps: \((-3) \cdot x = -3x\).

Key Concepts

Cube Root

Understanding the cube root is essential to solving problems like the one we have here. The cube root of a number is a special value such that, when you multiply it by itself twice more, you get the original number back. In mathematical terms, if you have a number \(a\), then \(a^{1/3}\) is the cube root of \(a\). In our exercise, we solve \((-27)^{1/3}\), where

• The cube root of \(-27\) is \(-3\). Why? Because \((-3) \times (-3) \times (-3) = -27\).

Whenever you encounter negatives in roots, consider if the root order permits negatives without creating complex numbers.

Exponentiation Rules

The power or exponentiation rules are like the alphabet of algebra. They help you unscramble complex algebraic terms into simpler forms. First, understand that when you apply an exponent to a product, such as

• \((a \, b)^n = a^n \cdot b^n\).

This means each factor in the product is raised to the power separately, which was used in our step by step solution. The expression \((-27 x^{3})^{1/3}\) became \((-27)^{1/3} \cdot (x^3)^{1/3}\). Exponentiation rules also include the concept of power of a power; \((a^m)^n = a^{m \cdot n}\), simplifying expressions like \((x^3)^{1/3}\) as shown earlier, leading you to \(x^{1}\), or just \(x\). With these rules, unwinding exponents becomes an intuitive process.

Simplifying Algebraic Expressions

Algebra simplification is about making expressions more manageable. This often involves breaking down complex parts into simpler units, and then combining them to achieve a simplified form. In our exercise:

• Take each element separately (for example, -27 and \(x^3\)).
• Use relevant operations, like finding cube roots in the first case, and splitting powers over a product in the latter.

Simplifying means finding the most basic form of an expression without changing its value. Step by step, we were able to transform the initial complex expression into a simple product, \(-3x\). By practicing these steps, handling algebraic expressions turns less daunting.

Negative Numbers in Algebra

Handling negative numbers in algebra is a unique yet frequent task that algebra students must get comfortable with. Here are some essentials:

• Negative numbers have peculiar behaviors, especially under roots and powers.
• In the context of cube roots; only odd roots of negatives are straightforward as they still belong to real numbers. For instance, \((-27)^{1/3}\) is \(-3\), not a complex number, because three \((-3)\) multiplied gives \(-27\).
• Negative values affect the sign of the product: remember \((-a) \times b = -ab\). Always pay attention to signs; they could change your results dramatically in simplifications.

Through carefully attending to these principles, negative quantities pose no sudden surprises.