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Chapter 10: Problem 23

Simplify completely. $$\sqrt[3]{-\frac{54}{2}}$$

Short Answer

Expert verified
The simplified expression is \(\sqrt[3]{-\frac{54}{2}} = -3\).

Step by step solution

01

1. Simplify the fraction inside the parentheses

To do this, divide -54 by 2, which results in -27: \[ \sqrt[3]{-\frac{54}{2}} = \sqrt[3]{-27} \]
02

2. Find the cube root

Now, find the cube root of -27, which means finding a number that when multiplied by itself three times gives -27. This number is -3: \[ \sqrt[3]{-27} = -3 \]
03

3. Write the final simplified expression

The cube root of \(-\frac{54}{2}\) simplifies to -3: \[ \sqrt[3]{-\frac{54}{2}} = -3 \]

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Key Concepts

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

Simplifying Fractions
Fractions sometimes look complicated, but they can be made simpler by dividing both the numerator and the denominator by their greatest common divisor (GCD). In the exercise provided, the fraction is \(-\frac{54}{2}\). To simplify it, consider:
  • The numerator is -54, and the denominator is 2.
  • The GCD of 54 and 2 is 2.
By dividing both the top and bottom by 2, the fraction simplifies to \(-27\). This process reduces the fraction to its simplest form, making it easier to work with in further calculations.
Note: The negative sign stays with the numerator or the fraction as a whole, as it indicates the negative nature of the entire fraction.
Radical Expressions
A radical expression involves roots, like square roots or cube roots. In this exercise, we deal with the cube root, represented as \(\sqrt[3]{}\). The cube root of a number \(x\) finds a value \(y\) such that \(y^3 = x\).Here, we are looking at \(\sqrt[3]{-27}\). Breaking it down:
  • -27 is the number inside the radical sign.
  • The job is to find a number \(y\), so that when it's multiplied by itself three times (\(y \times y \times y\)), it equals -27.
Finding this can involve trial and error or knowledge of elementary cube numbers. Ultimately, the cube root of -27 is -3, since \((-3) \times (-3) \times (-3) = -27\).
Radical expressions are simplified by resolving these types of roots straightforwardly and correctly.
Negative Numbers
Handling negative numbers requires an understanding of how they interact in mathematical operations. In this exercise, we're dealing with the cube root of a negative number, -27. Negative numbers have a unique behavior:
  • When multiplied an odd number of times, the result remains negative.
  • For example, \((-3) \times (-3) \times (-3) = -27\), which shows why the cube root of -27 is -3.
This concept helps when simplifying expressions involving negative bases in odd-root radicals.
Don't be daunted by the negative sign; just remember its properties in multiplication, and your calculations will be on the right track.

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Most popular questions from this chapter

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