Problem 87 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( a \).

Step 1: Apply the Product Rule of Exponents

When multiplying exponents with the same base, you add the exponents. Start by simplifying the numerator of the expression. Given: \( a^{3/4} \cdot a^{3/4} \), this becomes \( a^{3/4 + 3/4} = a^{6/4} \).

Step 2: Simplify the Fraction in the Exponent

The expression \( a^{6/4} \) can be simplified further. Since \( 6/4 = 3/2 \), the expression becomes \( a^{3/2} \).

Step 3: Apply the Quotient Rule of Exponents

When dividing exponents with the same base, you subtract the exponents. Use this rule to simplify the fraction: \( \frac{a^{3/2}}{a^{1/2}} \) becomes \( a^{3/2 - 1/2} = a^{2/2} = a^1 \).

Step 4: Express without Exponents

Since \( a^1 = a \), the expression does not need to be altered further.

Key Concepts

Product Rule of Exponents

The product rule of exponents is a fundamental principle that helps simplify expressions where the same base is raised to multiple exponents. When you multiply two or more terms with the same base, you simply add the exponents together. This rule makes dealing with exponential expressions easier and tidier.For example, if you have an expression like \(a^{m} \times a^{n}\), the product rule tells us this is equal to \(a^{m+n}\). Let's see how this works in our exercise:

• We start with \(a^{3/4} \cdot a^{3/4}\).
• Since the bases are the same, we add the exponents: \(3/4 + 3/4\).
• This results in \(a^{6/4}\).

By using the product rule, we have successfully combined the two exponents into one, making the expression neater and further simplifying the subsequent steps.

Quotient Rule of Exponents

The quotient rule of exponents is used when you are dividing terms with the same base. Instead of multiplying, you subtract the exponents from each other. This allows you to simplify expressions that involve division.If you encounter an expression like \(\frac{a^{m}}{a^{n}}\), the quotient rule helps us simplify this to \(a^{m-n}\). This rule was instrumental in the exercise as follows:

• We had a division of \(a^{3/2}\) by \(a^{1/2}\).
• Subtract the exponent in the denominator from the exponent in the numerator: \(3/2 - 1/2\).
• This results in \(a^{2/2}\), which simplifies further to \(a^1\).

By applying the quotient rule, the expression is reduced to a simpler form, making it more manageable to interpret and use.

Simplifying Expressions

Simplifying expressions is a process to make expressions easier to understand and work with by reducing them to their simplest form. In the context of exponents, this often involves applying the product and quotient rules to eliminate any unnecessary complexity.In our example, we simplified from a complex expression involving fractional exponents to a much simpler form:

• Initially, we had \(\frac{a^{3/4} \cdot a^{3/4}}{a^{1/2}}\).
• Using the product rule, this became \(a^{6/4}\).
• Simplifying \(6/4\) to \(3/2\), we simplified further to \(a^{3/2}\).
• Then, with the quotient rule, dividing by \(a^{1/2}\) led us to \(a^{1}\).
• Finally, since \(a^1\) is simply \(a\), the expression was completely simplified without changing its core meaning.

By gradually applying these rules and simplifying fractions along the way, we managed to express a complex fraction of exponents as just a single variable. This is the essence of simplifying expressions, making them as clear and straightforward as possible.