Problem 26 - Algebra 2

Question

Evaluate each expression. $$ 216^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

6

Step 1: Understand the Expression

The expression given is \( 216^{\frac{1}{3}} \). This is a cube root expression, where you are required to find the number that, when cubed, results in 216.

Step 2: Identify Perfect Cubes

Identify perfect cube numbers around 216. Note that \( 6^3 = 216 \), indicating that the cube root of 216 is 6.

Step 3: Apply the Cube Root Property

Evaluate \( 216^{\frac{1}{3}} \) by using the property of cube roots: \( a^{\frac{1}{3}} \) is equal to the cube root of \( a \). Thus, \( 216^{\frac{1}{3}} = 6 \).

Key Concepts

Perfect Cubes

A perfect cube is a number that can be expressed as the cube of an integer. It means that a number, when multiplied by itself three times, results in this perfect cube. For instance, 216 is a perfect cube because \(6 \times 6 \times 6 = 216\). Recognizing perfect cubes can simplify calculations with cube roots significantly. When faced with a number like 216, you should first check if it is a perfect cube. This involves identifying which integer, when cubed, equals the target number. In this case, 6 cubed gives us 216, confirming it is indeed a perfect cube.

• Example 1: \(3^3 = 27\). Therefore, 27 is a perfect cube.
• Example 2: \(5^3 = 125\). So, 125 is a perfect cube as well.

Recognizing perfect cubes quickly is a useful skill, especially for problems involving cube roots.

Exponents

Exponents are a mathematical notation that indicate the number of times a number, the base, is multiplied by itself. In the expression \(a^b\), \(a\) is the base, and \(b\) is the exponent. In relation to cube roots,

• An exponent of 1/3 indicates a cube root.
• Exponents are used to express very large or very small numbers compactly.

In the problem \(216^{\frac{1}{3}}\), the exponent \(\frac{1}{3}\) represents the cube root of 216. Thus, to evaluate any cube root, you apply the principle \(a^{\frac{1}{3}} = \sqrt[3]{a}\). This relationship between exponents and roots allows for straightforward calculation of root expressions using familiar math rules. Practice identifying and manipulating exponents to reinforce your understanding.

Root Expressions

Root expressions are another form of mathematical notation used to denote roots of a number. The notation \(\sqrt[n]{a}\) indicates the "n"th root of a number. Specifically, when \(n = 3\), it refers to a cube root, which is what the original problem describes.
To solve cube root expressions like \(216^{\frac{1}{3}}\), understand that:

• The notation \(\sqrt[3]{216}\) is equivalent to \(216^{\frac{1}{3}}\).
• Cube roots return a value that, when cubed, results in the original number.

These terms simplify and tidy up calculations, especially when dealing with large numbers. A root expression offers clarity and precision in mathematical solutions, making it easier to manipulate and solve equations involving roots. Always approach roots with a recognition of their power to bring a multi-factor number down to its base factor level.