Problem 23 - Intermediate Algebra

Question

Evaluate each expression. See Example 1. $$ (-216)^{1 / 3} $$

Step-by-Step Solution

Answer

The result is -6.

Step 1: Understanding the Expression

The expression \((-216)^{1/3}\) means we need to find the cube root of -216. This is equivalent to finding a number that, when multiplied by itself three times, results in -216.

Step 2: Solving for the Cube Root

Identify the cube root of -216. Since we are working with a negative number, we need to find a negative number that satisfies this condition. The number -6 satisfies, as \((-6) \times (-6) \times (-6) = -216\).

Step 3: Verification

To verify the result, re-calculate: \((-6) \times (-6) = 36\) and then \(36 \times (-6) = -216\). Hence the cube root of -216 is indeed -6.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front of them.

These numbers can seem a bit tricky at first, but they are just as important as positive numbers in mathematics. When you multiply negative numbers, the product can either be positive or negative depending on how many negative numbers are being multiplied together.

Here are some important points to remember about negative numbers:

• When you multiply two negative numbers, the result is positive. For example, \((-2) \times (-3) = 6\).
• When you multiply a positive number by a negative number, the result is negative. For example, \(2 \times (-3) = -6\).
• Negative numbers are used in various real-life scenarios, such as measuring temperature below zero or representing debts in financial accounts.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In mathematical terms, an exponent is written as a small number to the upper right of a base number. For example, in the expression \(3^4\), the number 3 is the base and 4 is the exponent which means \(3\times 3\times 3\times 3\).

Here are a few key points to understand about exponents:

• An exponent of 1 means the number remains the same. For instance, \(5^1 = 5\).
• An exponent of 0 means any base (except 0) raised to this power equals 1, such as \(5^0 = 1\).
• Fractional exponents, like \(x^{1/n}\), denote the nth root of x. So, \((-216)^{1/3}\) implies the cube root of -216.

Radicals

Radicals involve finding the root of numbers, with symbols like the square root (√) and cube root (∛). They help in simplifying expressions involving roots of numbers.

Radicals are closely related to exponents, specifically fractional exponents. For instance, \(\sqrt{x}\) can be written as \(x^{1/2}\), and the cube root of x, \(\sqrt[3]{x}\), can be expressed as \(x^{1/3}\).

Understanding radicals includes:

• The square root of x is a number that, when squared, gives x. For example, \(\sqrt{4} = 2\) because \(2\times 2 = 4\).
• The cube root is similar: it's a number that, when used in a three-way product, results in the original number, such as \(\sqrt[3]{8} = 2\) because \(2\times 2\times 2 = 8\).
• Radicals involving negative numbers mean finding negative roots since a negative number times itself three times gives a negative result, like in \((-6) \times (-6) \times (-6) = -216\).