Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 27^{\frac{4}{3}} $$

Step-by-Step Solution

Answer

The expression \( 27^{\frac{4}{3}} \) simplifies to 81.

Step 1: Understanding the Problem

We have the expression \( 27^{\frac{4}{3}} \) and we need to express it as a rational number in its simplest form.

Step 2: Rewriting the Base Number

Notice that 27 can be rewritten as \( 3^3 \). Therefore, \( 27^{\frac{4}{3}} \) can be rewritten as \((3^3)^{\frac{4}{3}}\).

Step 3: Simplifying the Exponentiation

When raising a power to a power, we multiply the exponents: \((3^3)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}}.\)

Step 4: Calculating the Product of Exponents

Multiply the exponents: \(3 \times \frac{4}{3} = 4.\)Thus, the expression becomes \(3^4\).

Step 5: Calculating the Final Value

Now we calculate \(3^4\): \(3^4 = 3 \times 3 \times 3 \times 3 = 81.\) Therefore, \( 27^{\frac{4}{3}} = 81 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value, making them easier to understand and work with. When working with expressions like \( 27^{\frac{4}{3}} \), simplification often entails dealing with exponents to make calculations more manageable.

To simplify \( 27^{\frac{4}{3}} \), we first recognize that 27 can be expressed as a base raised to an exponent, specifically \( 27 = 3^3 \). This change helps to clarify the expression and set us up for applying exponentiation rules. From here, we rewrite the expression by raising \( 3^3 \) to the power \( \frac{4}{3} \), leading to \( (3^3)^{\frac{4}{3}} \).

As a rule, when you raise a power by another exponent, you multiply these exponents together. This simplification step ultimately transforms the expression into \( 3^{3 \times \frac{4}{3}} \), which further simplifies to \( 3^4 \). Simplified to its base, \( 3^4 \), the complexity is reduced even though the original value remains the same.

Exponentiation

Exponentiation is a mathematical operation involving raising a base to a power or exponent. This operation is frequently utilized when dealing with expressions where a number is multiplied by itself several times. In the problem \( 27^{\frac{4}{3}} \), exponentiation plays a key role in both manipulating and simplifying the expression.

Using rules of exponents, specifically the power of a power property, allows us to simplify expressions like \( (3^3)^{\frac{4}{3}} \). The property states that when you take a power of a power, you multiply the exponents: \( b^{m^n} = b^{m \times n} \).

Thus, the step is to calculate \( 3^{3 \times \frac{4}{3}} \). By multiplying \( 3 \) and \( \frac{4}{3} \), you get 4, transforming the expression to \( 3^4 \). Such a simplification demonstrates the elegance of using exponentiation rules to reduce complex expressions to more workable formats, allowing us to easily find the numerical result like 81 in this case.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). They play a significant role in various mathematical processes and in simplifying expressions like \( 27^{\frac{4}{3}} \).

The exponent \( \frac{4}{3} \) in \( 27^{\frac{4}{3}} \) is a rational number, representing a division in the power applied to 27. This signifies taking a cube root first (denominator 3), and then raising the result to the fourth power (numerator 4).

Understanding rational exponents is crucial for comprehending how operations alter numbers and expressions. Here, \( 27^{\frac{4}{3}} \) invites examining the relationship between roots and powers, showing how complex expressions yield simple rational results: first simplifying to \( 3^4 \), and finally contributing to a result of 81, a testament to the power of evaluating rational functions effectively.