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Chapter 0: Problem 29

Simplify each expression. a. \(27^{1 / 3}\) b. \((-27)^{1 / 3}\) c. \(-27^{1 / 3}\)

Short Answer

Expert verified
a. 3, b. -3, c. -3

Step by step solution

01

Simplify a. \(27^{1/3}\)

To simplify \(27^{1/3}\), recognize that \(27\) is a perfect cube. Specifically, \(27 = 3^3\). Hence, \(27^{1/3} = (3^3)^{1/3} = 3^{3\cdot(1/3)} = 3\). Therefore, \(27^{1/3} = 3\).
02

Simplify b. \((-27)^{1/3}\)

To simplify \((-27)^{1/3}\), recognize that \(-27\) is also a perfect cube of a negative number. Specifically, \(-27 = (-3)^3\). Hence, \((-27)^{1/3} = ((-3)^3)^{1/3} = (-3)^{3\cdot(1/3)} = -3\). Therefore, \((-27)^{1/3} = -3\).
03

Simplify c. \(-27^{1/3}\)

This expression means taking the cube root of \(27\) first and then applying the negative sign. From Step 1, we know that \(27^{1/3} = 3\). So, \(-27^{1/3} = -3\). Therefore, \(-27^{1/3} = -3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When we talk about simplifying expressions, it often involves breaking down complex mathematical phrases into their simplest form. Simplification can make it easier to work with and understand the core idea. If you see an expression like \(27^{1/3}\), you can simplify it by recognizing it as a perfect cube. Here are some tried and tested steps to simplify such expressions:
  • Identify the base number and exponent.
  • Recognize if the base is a perfect cube. A perfect cube is a number like 8, 27, or 64, which can be written as \(2^3, 3^3, 4^3\).
  • Rewrite the base using its cube form. For instance, \(27 = 3^3\).
  • Apply the exponent rule for powers and then simplify. For example, \((3^3)^{1/3} = 3^{3 \times (1/3)} = 3\).
Following these steps ensures that you break the problem down and understand each component clearly, leading to a simpler final answer.
Perfect Cube
A perfect cube is a number that can be expressed as the product of an integer multiplied by itself three times. In simpler words, a perfect cube results from raising an integer to the power of three. For example, \(27\) is a perfect cube because it is equal to \(3^3\). There are a few things to remember about perfect cubes:
  • Both positive and negative numbers can be perfect cubes. For example, \(8 = 2^3\) and \(-27 = (-3)^3\).
  • Perfect cubes are useful for simplifying expressions that involve cube roots.
  • Recognizing perfect cubes can simplify operations, such as finding cube roots quickly.
Understanding perfect cubes helps you break down and simplify complex expressions, as we've seen in the step-by-step solutions provided.
Negative Numbers
Negative numbers can sometimes be tricky, especially when combined with exponents and roots. When working with negative numbers, here are some key points to remember:
  • The cube root of a negative number is also negative. For example, the cube root of \(-27\) is \(-3\) because \((-3)^3 = -27\).
  • When dealing with expressions like \(-27^{1/3}\), it's important to apply the negative sign after taking the cube root of \(27\). This means \(-27^{1/3}\) simplifies to \(-3\) because \(27^{1/3} = 3\).
  • Negative exponents and roots need careful attention to avoid mistakes. Always perform operations with parentheses when necessary to maintain clarity.
Keeping these principles in mind makes handling negative numbers more straightforward, especially in expressions and equations involving roots and powers.

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