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Chapter 10: Problem 63

Find each root, if possible. $$-\sqrt[3]{-27}$$

Short Answer

Expert verified
The final answer for the expression \(-\sqrt[3]{-27}\) is \(3\), as this simplifies to \(-\left(-27\right)^{\frac{1}{3}} = -(-3)\).

Step by step solution

01

Understand the problem

We need to find the cube root of a negative number, specifically \(-\sqrt[3]{-27}\).
02

Apply the cube root

The cube root of a number \(x\) can be represented as \(x^{\frac{1}{3}}\), so we rewrite our expression: \(-\left(-27\right)^{\frac{1}{3}}\).
03

Evaluate the expression

Now we can calculate the cube root of -27: \(-\left(-27\right)^{\frac{1}{3}} = -\left(-3\cdot -3 \cdot -3\right)^{\frac{1}{3}} = -(-3)\).
04

Final Answer

The final answer is 3. The cube root of -27 is -3, and the expression evaluates to 3.

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They are found to the left of zero on the number line and often represent opposite directions, losses, or decreasing values. For example, owing $10 is represented by -10 because it indicates that you have less than zero cash. Understanding negative numbers is crucial in algebra and other branches of mathematics.
  • Negative numbers can be combined with positive numbers or other negative numbers using standard arithmetic operations.
  • Multiplying or dividing a negative number by a positive number will result in a negative number.
  • Multiplying or dividing two negative numbers results in a positive number.
A unique property of negative numbers is their behavior when raised to an odd or even power. When a negative number is raised to an odd power, the result is negative. However, if raised to an even power, the result is positive.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. These symbols represent numbers and express generalized operations or formulas.
Algebra allows us to create equations that we can solve with various methods. A key component in algebra is understanding and solving for unknown variables.
  • One of the basic laws of algebra involves the order of operations, commonly remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Another crucial aspect is simplifying expressions, which often involves combining like terms or factoring.
In our case with cube roots, algebra provides a framework for solving expressions such as \(-\sqrt[3]{-27}\). Using algebraic techniques, we interpret and find solutions to expressions involving roots and exponents.
Roots of Negative Numbers
Roots, especially cube roots, can involve negative numbers. A root is a value that, when multiplied by itself a certain number of times, results in the original number.
For cube roots, this means finding a number that multiplied by itself twice results in the original number.
  • The cube root of a negative number is negative. This is because multiplying three negative numbers gives a negative result.
  • Unlike square roots, which are not real when negative numbers are involved, cube roots of negative numbers yield real values.
In the exercise provided, the term \(-27\) indicates we need the cube root of \(-27\), which is \(-3\) because \(-3 \times -3 \times -3 = -27\). The negative sign outside the cube root \(-\sqrt[3]{-27}\) flips the result, finalizing the solution as 3.

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