Problem 85 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \frac{3^{4 / 3} 3^{1 / 3}}{3^{2 / 3}} $$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Apply the Exponent Addition Rule

The first step is to simplify the expression in the numerator by using the property of exponents that states \( a^m \times a^n = a^{m+n} \). Therefore, in the numerator, we have \( 3^{4/3} \times 3^{1/3} = 3^{4/3 + 1/3} = 3^{5/3} \).

Step 2: Simplify the Fraction Using the Exponent Subtraction Rule

Now, simplify the fraction using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to \( \frac{3^{5/3}}{3^{2/3}} = 3^{5/3 - 2/3} = 3^{3/3} \).

Step 3: Simplify the Exponent

Finally, simplify the exponent: \( 3^{3/3} = 3^1 = 3 \). Thus, the expression simplifies to 3.

Key Concepts

Exponent Rules

Exponent rules are mathematical principles that govern how we manipulate powers in equations. They help us simplify complex expressions involving exponents. A key rule, known as the "Exponent Addition Rule," states that when multiplying two powers with the same base, you can simply add the exponents.

For example: When you have an expression like \( a^m \times a^n \), you can rewrite it as \( a^{m+n} \). It allows you to change multiplication into addition, making calculations simpler and more efficient.

• Example: \( 3^{4/3} \times 3^{1/3} = 3^{(4/3)+(1/3)} = 3^{5/3} \)

Another important rule is the "Exponent Subtraction Rule." This is useful for dividing powers with the same base, where you subtract the exponent of the denominator from the exponent of the numerator.

• Example: \( \frac{a^m}{a^n} = a^{m-n} \)

Positive Exponents

Positive exponents denote the number of times a base is multiplied by itself. For example, with a positive integer exponent like \( a^3 \), it means \( a \times a \times a \). Positive exponents simplify the understanding of multiplication of numbers, powering a number positively and are always straightforward to work with.

Understanding positive exponents also means knowing that they ensure the result remains straightforward, leading to a positive number or expression. Reducing an equation to only have positive exponents is often a goal since it simplifies expressions and avoids using negative exponents which could complicate situations further.

In practice, you avoid negative exponents by ensuring all exponents in the final expression are positive, which often involves rearranging expressions using exponent rules. This is seen clearly in how expressions are simplified to ensure that denominators are free of negative exponents.

Simplifying Fractions

Simplifying fractions with exponents often involves using exponent rules, specifically focusing on making sure the expressions are easy to interpret and calculate. One crucial step in simplifying fractions of exponential expressions is to apply the Exponent Subtraction Rule correctly. It involves moving from a complex fraction involving powers to a more manageable single power or integer.

• For instance, \( \frac{3^{5/3}}{3^{2/3}} = 3^{(5/3)-(2/3)} = 3^{3/3} = 3^1 \)

Here, the fraction was simplified to a form with a positive exponent. Simplifying fractions ensures that we end up with the simplest form possible, which for exponential expressions, often means removing fractions or complex parts of powers to make calculations much easier. The goal is always to render the expression as simple and clean as possible, which often leads directly to the simplest integer or a straightforward exponential form.