Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\sqrt[5]{3^{7}} \cdot \sqrt[5]{3^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Rewrite the Roots as Exponential Expressions

Rewrite \( \sqrt[5]{3^{7}} \) and \( \sqrt[5]{3^{3}} \) as \( 3^{7/5} \) and \( 3^{3/5} \) respectively. This shows that taking the 5th root is the same as raising the number to the \(1/5th\) power.

Step 2: Multiply the Expressions

Since the bases are identical, you add the exponents together when multiplying. This means that \( 3^{7/5} \cdot 3^{3/5} \) simplifies to \( 3^{(7/5 + 3/5)} \).

Step 3: Simplify the Final Expression

Now, simply perform the addition in the exponent to find \( 3^{(7/5 + 3/5)} = 3^{2} = 9 \).

Key Concepts

Exponential Expressions

Understanding exponential expressions is essential for simplifying radical expressions. An exponential expression is written in the form of a base raised to the power of an exponent. For instance, in the expression \(3^{2}\), 3 is the base, and 2 is the exponent, which tells us how many times the base is multiplied by itself. Thus, \(3^{2}\) is the same as \(3 \times 3 = 9\).

Exponential expressions can also have fractions as exponents, like \(3^{7/5}\), which represents the 5th root of \(3^7\). Fractional exponents are particularly useful when dealing with radical expressions, which typically involve roots. The top number of the fraction indicates the power, while the bottom number indicates the type of root. Hence, in simplifying, we can convert between root and fractional exponent forms to make the process easier.

Properties of Exponents

The properties of exponents play a pivotal role in simplifying expressions. Three fundamental rules are often applied:

• Product Rule: When multiplying two expressions with the same base, add the exponents (e.g., \(x^{m} \times x^{n} = x^{m+n}\)).
• Quotient Rule: When dividing two expressions with the same base, subtract the exponents (e.g., \(x^{m} / x^{n} = x^{m-n}\)).
• Power Rule: When raising an exponent to another power, multiply the exponents (e.g., \( (x^{m})^{n} = x^{mn} \)).

Using these rules, you can simplify complex expressions efficiently. For instance, the given exercise multiplies two expressions with the same base of 3 but with different exponents. By applying the Product Rule, we add the exponents: \(3^{7/5} \times 3^{3/5} = 3^{7/5 + 3/5}\).

Radical to Exponential Form

Converting a radical to exponential form is a standard technique to simplify radical expressions. A radical, such as \(\sqrt[n]{a}\), can be written as \(a^{1/n}\) in exponential form. Here, 'n' is the root and 'a' is the radicand, the number under the root sign. This conversion is particularly useful because it allows us to employ the properties of exponents.

In the exercise, \(\sqrt[5]{3^{7}} \cdot \sqrt[5]{3^{3}}\) is rewritten as \(3^{7/5} \cdot 3^{3/5}\). Then, using our knowledge of properties of exponents, we perform the addition in the exponent to end up with \(3^{2}\), which simplifies further to 9. This process makes complex roots much easier to handle and helps in accurately simplifying expressions.