Problem 17 - Algebra 2

Question

Simplify. $$ \sqrt[3]{-27} $$

Step-by-Step Solution

Answer

The cube root of \(-27\) is \(-3\).

Step 1: Understanding the Cube Root

The cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). This means we are looking for a number that, when raised to the third power, gives us \(-27\).

Step 2: Identify the Cube Root

The number that, when cubed, equals \(-27\) is \(-3\). This is because \((-3) \times (-3) \times (-3) = -27\).

Step 3: Verify the Answer

Let's verify that our solution is correct. Raise \(-3\) to the power of 3: \((-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27\). This confirms that \(-3\) is indeed the cube root of \(-27\).

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we're working towards finding the simplest form of a radical expression. Radicals involve roots, such as square roots, cube roots, or higher.
To simplify a radical, particularly a cube root, you need to determine what number multiplied by itself a certain number of times results in the value under the radical sign. In this case, with the cube root of \(-27\), we're searching for a number that when cubed equals \(-27\).
Here's how simplifying a radical cube root generally works:

• Identify the number under the radical and consider its factors.
• Look for a perfect cube among those factors.
• Factor it out, simplifying the expression.

With \(-27\), we realize that \(-3\) is a perfect cube, because \((-3) imes (-3) imes (-3) = -27\).
Simplifying cube roots can appear daunting, particularly with negative numbers, but identifying a perfect cube makes it straightforward.

Negative Numbers

Understanding negative numbers is crucial in working with radicals, especially when simplifying cube roots. A negative number is any number less than zero. The unique trait about negative numbers is their behavior when raised to odd or even powers.
For example, when you raise a negative number to an odd power, like three for cube roots, the result is still negative. Consider \((-3)^3\); when multiplied consecutively as \((-3)\times (-3)\times (-3) = 9 \times (-3) = -27\), the answer remains negative.
By contrast, raising a negative number to an even power results in a positive number. For example, \((-2)^2 = 4\).
Remembering this behavior:

• Odd power - negative result for negative base.
• Even power - positive result for negative base.

For cube roots, thus, a negative under the radical indicates a negative root.

Exponents

Exponents are a fundamental component of simplifying expressions like cube roots. They reflect how many times to multiply a number by itself.
In algebra, the exponent known as the cube is three. This means, when dealing with a cube root like \(\sqrt[3]{-27}\), you are essentially reversing the process of cubing a number. You're looking for a number which, when raised to the power of three, yields \(-27\).
Here's a recap:

• An exponent describes repeated multiplication.
• In \((-3)^3 = -27\), the exponent is 3, indicating multiplying \((-3\)) three times.

When simplifying cube roots:- Understand the role of exponents.- Identify the number that fits the result under the cube root by checking potential cubes (like \(3^3\) or \((-3)^3\)).
This understanding of exponents is crucial in correctly solving expressions involving cube roots.