Problem 540 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((-216)^{\frac{1}{3}}\) (b) \(-216^{\frac{1}{3}}\) (c) \((216)^{-\frac{1}{3}}\)

Step-by-Step Solution

Answer

a) \(-6\), b) \(-6\), c) \(\frac{1}{6}\)

Step 1: Simplify \( (-216)^{\frac{1}{3}} \)

First, we need to find the cube root of -216. The cube root of -216 is -6 because \( (-6)^3 = -216 \). Therefore, \( (-216)^{\frac{1}{3}} = -6 \).

Step 2: Simplify \( -216^{\frac{1}{3}} \)

Here, the exponent applies only to 216, not the negative sign. We first find the cube root of 216, which is 6 because \( 6^3 = 216 \). So, applying the negative sign: \( -216^{\frac{1}{3}} = -6 \).

Step 3: Simplify \( (216)^{-\frac{1}{3}} \)

A negative exponent indicates a reciprocal. Therefore, we have \( (216)^{-\frac{1}{3}} = \frac{1}{(216)^{\frac{1}{3}}} \). The cube root of 216 is 6. Thus, \( (216)^{-\frac{1}{3}} = \frac{1}{6} \).

Key Concepts

Cube Roots

A cube root is a value that produces a given number when cubed. In other words, if you have a number and you're told that it's the cube root of another number, it means that raising it to the power of three will get you the original number. For example, since \(6^3 = 216\), the cube root of 216 is 6. Cube roots can also be taken of negative numbers. This is because the cube of a negative number is also negative. For example, since \( (-6)^3 = -216 \), the cube root of -216 is -6. This is reflected in problem parts (a) and (b) from the exercise.

Negative Exponents

Negative exponents signal that we should take the reciprocal of the base with the positive exponent. For example, \(3^{-2}\) means \(\frac{1}{3^2}\). This makes negative exponents very useful for simplifying expressions involving division and fractions. Here's a breakdown of part (c) in the exercise: For \(216^{-\frac{1}{3}}\), we see the negative exponent \(- \); hence we take the reciprocal first. We get \( \frac{1}{216^{\frac{1}{3}}} \). We know that the cube root of 216 is 6. Hence, \( 216^{-\frac{1}{3}} = \frac{1}{6} \). This conversion is a crucial step in simplifying expressions that involve negative exponents.

Simplifying Expressions

Simplifying expressions means making them easier to understand or solving in their simplest form. It often involves operations like combining like terms, factoring, and executing arithmetic operations including using exponents and roots.

• Consider part (a): \(-216^{\frac{1}{3}}\), finding the cube root simplifies it to -6.
• For part (b): \(-216^{\frac{1}{3}}\), the cube root of 216 is 6, then applying the negative sign gives -6.
• And in part (c), \(216^{-\frac{1}{3}} \), we use the rule of negative exponents to simplify it first to \(\frac{1}{216^{\frac{1}{3}}}\), which simplifies to \(\frac{1}{6} \).

Simplifying expressions efficiently often requires understanding and applying key mathematical rules correctly. These steps are essential for precisely tackling the original problem.