Problem 74 - Elementary and Intermediate Algebra

Question

Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{-27} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{-27} \) is \(-3\).

Step 1: Identify the Cube Root Problem

The problem in front of us is to simplify the cube root \( \sqrt[3]{-27} \). This means we need to find a number which, when multiplied by itself three times, will equal \(-27\).

Step 2: Express the Cube Root Mathematically

We can express the cube root of \(-27\) mathematically as \( \sqrt[3]{-27} = x \). This implies that \(x^3 = -27\).

Step 3: Identify the Principal Cube Root

The cube root of a negative number will also be negative. Since \(27\) is \(3 \times 3 \times 3\), the cube root of \(-27\) will be \(-3\) because \((-3)^3 = -27\).

Step 4: Conclusion – Simplified Cube Root

There are no further steps needed for simplification. Thus, the simplified form of \( \sqrt[3]{-27} \) is \(-3\).

Key Concepts

Understanding Negative Numbers in Cube Roots

When dealing with cube roots, you may encounter negative numbers. A cube root determines which number can be multiplied by itself three times to reach a given number, which can also be negative. Unlike square roots, which cannot handle negatives under the radical without involving imaginary numbers, cube roots can manage negative inputs easily.

This is because negative numbers behave differently from positive numbers when multiplied an odd number of times, like three. Multiplying a negative number by itself an odd number of times will yield a negative result. For example,

• If you cube (-1), you get (-1) × (-1) × (-1) = -1.
• Similarly, (-3) cubed is (-3) × (-3) × (-3) = -27.

Thus, the cube root of (-27) is (-3), because when you multiply (-3) by itself three times, you indeed get (-27).

The Simplification Steps for Cube Roots

Simplifying the cube root, especially when a negative number is involved, can seem tricky at first. Nonetheless, by following a neat step-by-step process, it becomes simplified. Firstly, identify the cube root problem by expressing your number mathematically as \(\sqrt[3]{-27} = x\).

You need to solve \(x^3 = -27\). Here, finding x is crucial. Start by considering the elements of 27, which are3 x3 x3. Since you begin with (-27), you simply need to include the negative aspect by multiplying these numbers and making the final result negative. Therefore, the answer will be (-3), as \((-3) \times (-3) \times (-3) = -27\).

To summarize simplification:

• Recognize the cube root problem.
• Express it mathematically.
• Identify the elements involved.
• Utilize the properties of multiplication for negative numbers.
• Get to the simplified result.

Interpreting Mathematical Expressions

Mathematical expressions are combinations of symbols that express a mathematical concept or equation. The cube root symbol,\(\sqrt[3]{x}\), is a type of mathematical expression. It signifies finding a value that is multiplied by itself three times to yield x.

When interpreting expressions like \(\sqrt[3]{-27}\), it's important to understand what these symbols represent. The negative sign is part of the expression, so your solution must account for it. The expression says: "What number, when multiplied three times by itself, matches the negative original?"

Understanding expressions is also about breaking them down. View the components individually,

• the cube root symbol (\(\sqrt[3]{}\)),
• the negative number (\(-27\)).

Perform the operation to provide clarity. Recognize that mathematical expressions help convey complex ideas concisely and are essential in solving problems precisely and efficiently.