Problem 41 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[5]{\frac{3 x^{10}}{32}} $$

Step-by-Step Solution

Answer

\( \frac{x^2 \cdot \sqrt[5]{3}}{2} \) is the simplified form.

Step 1: Simplify the root of the fraction

Our task is to simplify the expression \( \sqrt[5]{\frac{3 x^{10}}{32}} \). Start by applying the fifth root to both the numerator and the denominator separately: \( \sqrt[5]{3 x^{10}} \) and \( \sqrt[5]{32} \).

Step 2: Simplify the fifth root of the numerator

Examine the numerator \( \sqrt[5]{3 x^{10}} \). To simplify, notice that \( x^{10} = (x^2)^5 \). Therefore, \( \sqrt[5]{3 x^{10}} = \sqrt[5]{3 (x^2)^5} = x^2 \cdot \sqrt[5]{3} \).

Step 3: Simplify the fifth root of the denominator

For the denominator, calculate \( \sqrt[5]{32} \). Recognize that 32 is a power of 2: \( 32 = 2^5 \). Therefore, \( \sqrt[5]{32} = 2 \).

Step 4: Combine simplified elements

Now, combine the results from Steps 2 and 3: \[ \frac{\sqrt[5]{3 x^{10}}}{\sqrt[5]{32}} = \frac{x^2 \cdot \sqrt[5]{3}}{2} \].

Step 5: Simplify the complete expression

The simplified expression is: \[ \frac{x^2 \cdot \sqrt[5]{3}}{2} \].

Key Concepts

Understanding Exponents

Exponents are powerful mathematical tools that show how many times a number, termed the base, is multiplied by itself. In our exercise, we encounter an exponent in the form of the expression \( x^{10} \). This tells us that the base \( x \) is multiplied by itself 10 times.

• For example, \( x^3 \) means \( x \times x \times x \).
• In another case, \( x^{0} \) equals 1, because any number raised to the power of zero is 1.
• Exponents provide a shorthand way of writing repeated multiplication, which helps simplify long expressions.

When simplifying expressions with exponents, like \( x^{10} \), remember we can rewrite them. \( x^{10} = (x^2)^5 \) implies that \( x^{10} \) is composed of \( x^2 \) multiplied by itself 5 times.
This property is useful for simplifying radical expressions, particularly when involving roots.

Determining the Fifth Root

The fifth root of a number is what you multiply by itself five times to get the original number. In the equation \( \sqrt[5]{\frac{3 x^{10}}{32}} \), the fifth root is a key element in simplifying the expression.

• For any number \( a \), the fifth root can be expressed as \( a^{1/5} \).
• Think of extracting roots as the opposite of using exponents. Where exponents show multiplication, roots show division.

To simplify our given expression, we break it down into considering both the numerator \( \sqrt[5]{3 x^{10}} \) and the denominator \( \sqrt[5]{32} \) separately. This makes the process easier and more manageable.
For instance, recognizing that \( 32 \) is equivalent to \( 2^5 \) allows us to simplify \( \sqrt[5]{32} \) directly to \( 2 \). Such simplifications streamline the process of dealing with large or seemingly complex numbers.

Numerator and Denominator Simplification

Simplifying either the numerator or the denominator of a fraction can often make solving the whole expression much easier. With the expression \( \sqrt[5]{\frac{3 x^{10}}{32}} \), doing this separately makes sense.

• Consider the numerator first: \( \sqrt[5]{3 x^{10}} \). Here, use the property of exponents, recognizing \( x^{10} \) as \( (x^2)^5 \), to pull \( x^2 \) out as a whole number. This step leaves us with \( x^2 \cdot \sqrt[5]{3} \).
• For the denominator, knowing \( 32 = 2^5 \), simplifies \( \sqrt[5]{32} \) to a neat \( 2 \).

When combined, these simplifications combine to an expression \( \frac{x^2 \cdot \sqrt[5]{3}}{2} \). Breaking the task into smaller parts often simplifies the process, giving clarity and making solution steps easier to follow.