Problem 18 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 32^{\frac{4}{3}} $$

Step-by-Step Solution

Answer

The simplest rational form of \(32^{\frac{4}{3}}\) is 16.

Step 1: Understand the Expression

The expression given is \(32^{\frac{4}{3}}\). This represents taking the 4th power of the cube root of 32.

Step 2: Find the Cube Root

Calculate the cube root of 32 by determining what number, when raised to the power of 3, equals 32. The cube root of 32 is 2, because \(2^5 = 32\).

Step 3: Raise to the Fourth Power

Since the cube root of 32 is 2, the next step is to raise this to the 4th power: \(2^4\). Compute this to find \(16\).

Step 4: Express as a Rational Number

The calculated value \(16\) is already a rational number, and it is in its simplest form.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is multiplied by itself a certain number of times. The number of times the base is multiplied is determined by the exponent.
In essence, when we write something like \(a^b\), \(a\) is our base and \(b\) is our exponent. This means that we are multiplying \(a\) by itself, \(b\) times. Exponents can be positive, zero, or negative, and they can also take on rational values, which involve fractions in the exponent. Rational exponents indicate roots and powers simultaneously.
For example in \(32^{\frac{4}{3}}\), the 3 in the denominator signifies a cube root and the 4 in the numerator means taking that result to the fourth power. In exponential terms, we separate the expression to \((32^{\frac{1}{3}})^4\), adapting it through root and power calculations.

Cube Roots

Cube roots are all about finding a number which, if multiplied by itself three times (cubed), gives the original number.
For example, the cube root of 27 is 3, since \(3 \times 3 \times 3 = 27\). The cube root is denoted using the radical symbol with a small three above it (\(\sqrt[3]{x}\)), or by expressing the power as a third like in \(x^{\frac{1}{3}}\).
When dealing with rational exponents, cube roots are quite common, especially when the fraction is in the form \(\frac{1}{n}\). In our exercise, when we find the cube root of 32, we ask: "What number, cubed, equals 32?" This number is 2 because \(2^3 = 8\), and upon another squaring, \(2^5 = 32\). It's a little tricky because it might take a reviewing of few smaller cubes to recognize the pattern.

Simplifying Expressions

Simplifying expressions involves reducing algebraic expressions into their simplest form. This often means eliminating all unnecessary parts, using arithmetic operations and combining like terms efficiently.
With rational exponents, simplifying involves a careful balance of breaking down roots and powers. Once the root has been simplified, you should then apply any remaining powers.
In the exercise with \(32^{\frac{4}{3}}\), the simplification began by finding the cube root first, simplified to 2. Afterward, raising this to the fourth power yielded \(2^4 = 16\). The result is already simplified into its most basic rational form. The goal of simplification is to have the expression as clear and straightforward as possible, which helps in further calculations or when trying to interpret outcomes.