Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 216^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplest form of \( 216^{\frac{1}{3}} \) is 6.

Step 1: Understanding the Expression

We need to express the given expression \( 216^{\frac{1}{3}} \) as a rational number in its simplest form. The expression involves a fractional exponent, which means we are dealing with a root. In particular, \( x^{\frac{1}{3}} \) represents the cube root of \( x \).

Step 2: Identify the Cube Root

Identify the number whose cube gives 216. In other words, find \( n \) such that \( n^3 = 216 \).

Step 3: Simplify the Cube Root

Check small integers to determine if their cube equals 216. Note that \( 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216 \). Therefore, \( 216^{\frac{1}{3}} = 6 \).

Step 4: Express the Result

Since \( 216^{\frac{1}{3}} = 6 \), the simplest rational form of the original expression is 6.

Key Concepts

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. In other words, a rational number is any number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).

• Examples are \( \frac{3}{4} \), \( -\frac{11}{5} \), or even \( 7 \) (which can be expressed as \( \frac{7}{1} \)).
• Rational numbers can be positive, negative, or zero.
• The decimal representation of rational numbers either terminates or repeats.

The power expression given in the exercise, \( 216^{\frac{1}{3}} \), ultimately results in a rational number because its value is equivalent to the whole number 6, which can be expressed as \( \frac{6}{1} \). Friendly reminder, a whole number is also considered a rational number!

Demystifying the Cube Root

The cube root of a number is another number that, when multiplied by itself twice more, equals the original number. Mathematically, if \( n^3 = x \), then \( n \) is the cube root of \( x \), which is represented by \( x^{\frac{1}{3}} \). Here's how it works:

• The cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
• The cube root of 27 is 3, as \( 3 \times 3 \times 3 = 27 \).
• For the exercise, finding the cube root of 216 involves identifying \( n \) such that \( n^3 = 216 \).

To find the cube root, experiment with numbers starting from 1, squaring and then cubing each until the match is found. In this case, \( 6^3 = 216 \), which means \( 6 \) is the cube root of 216. Therefore, \( 216^{\frac{1}{3}} = 6 \).

Expressing in the Simplest Form

The simplest form of a number refers to expressing it in the most reduced, efficient, and basic way possible. For rational numbers, simplification typically means reducing the fraction so that the greatest common divisor (GCD) of the numerator and denominator is 1.

• For integers, the simplest form means representing it as a basic whole number.
• In the case of the exercise, \( 216^{\frac{1}{3}} \) is simplified to \( 6 \), because no further reduction is possible, and \( 6/1 \) is already in simplest form.

Simplifying expressions often requires understanding and manipulating exponents and common factors to ensure the result is as uncomplicated as possible. In this instance, expressing \( 216^{\frac{1}{3}} \) resulted in the simplest rational number: \( 6 \).