Problem 86 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \frac{2^{5 / 6} 2^{1 / 3}}{2^{1 / 2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2^{2/3}\).

Step 1: Apply the Product of Powers Rule

The Product of Powers Rule states that when you multiply two powers with the same base, you add the exponents. We can apply this rule to the numerator: \(2^{5/6} \cdot 2^{1/3} = 2^{5/6 + 1/3}\). To add the exponents, convert \(1/3\) to \(2/6\). Therefore, \(5/6 + 2/6 = 7/6\). So, the numerator becomes \(2^{7/6}\).

Step 2: Use the Quotient of Powers Rule

The Quotient of Powers Rule states that when you divide two powers with the same base, you subtract the exponents. Here, we do \(2^{7/6} \div 2^{1/2}\). Subtract \(1/2\) from \(7/6\). Convert \(1/2\) to \(3/6\) to have a common denominator. Thus, \(7/6 - 3/6 = 4/6\). This simplifies to \(2^{4/6}\), which is \(2^{2/3}\).

Step 3: Simplify the Expression with Positive Exponents

As a final step, we ensure the expression has no negative exponents and is as simple as possible. The simplified expression is \(2^{2/3}\), which already has only positive exponents.

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental concept in algebra that simplifies expressions involving exponents. When you multiply two exponents with the same base, you simply add the exponents together. This rule makes it easier to handle expressions with powers, especially those with fractional exponents.

For example, in the expression \(2^{5/6} \cdot 2^{1/3}\), the base is the same (which is 2). To simplify, convert \(1/3\) into a fraction with the same denominator as \(5/6\). This is \(2/6\).

Now, you can add these fractions:

• \(5/6 + 2/6 = 7/6\)

The exponent on the base here becomes \(2^{7/6}\). This shows how following the Product of Powers Rule can tidy up an expression significantly, making it manageable and easier to use in further calculations.

Quotient of Powers Rule

The Quotient of Powers Rule helps simplify expressions when dividing exponents with the same base. According to this rule, you subtract the exponent of the divisor from the exponent of the dividend.

In our exercise, we have \(2^{7/6}\) divided by \(2^{1/2}\). The base in both parts of the fraction is the same, allowing us to apply this rule readily.

Firstly, make sure the exponents have a common denominator. Convert \(1/2\) into \(3/6\) to do this. Now subtract the exponent in the denominator from the one in the numerator:

• \(7/6 - 3/6 = 4/6\)

This simplifies further to \(2^{2/3}\). The Quotient of Powers Rule is a handy tool for streamlining fractions involving powers, dramatically reducing the complexity of problems.

Simplifying Expressions

Simplifying expressions is all about reducing them to their most manageable form and ensuring they follow certain mathematical rules, such as keeping all exponents positive.

After applying the Product and Quotient of Powers Rules, you might find powers with negative exponents or fractions. The goal is to reduce these so the expression remains simple and without negative exponents.

In our case, we started with a more complex fraction. But after applying these rules, we simplified it right down to \(2^{2/3}\). Let’s break down the importance of simplifying even more:

• It makes the expression easier to work with in future calculations.
• Simplified expressions reduce the potential for errors.
• They follow the general guidelines of mathematical elegance and efficiency.

By focusing on reducing expressions and ensuring they have positive exponents, you ensure clarity and simplicity, crucial aspects for effective math solutions.