Problem 82 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[10]{a^{5} b^{5}} $$

Step-by-Step Solution

Answer

\( a^{\frac{1}{2}} b^{\frac{1}{2}} \).

Step 1: Recognize the Radical as an Exponential Expression

The given expression \( \sqrt[10]{a^5 b^5} \) can be rewritten using rational exponents. The nth root of a number \( x \) is equivalent to raising \( x \) to the power of \( \frac{1}{n} \). Thus, \( \sqrt[10]{a^5 b^5} \) is \( (a^5 b^5)^{\frac{1}{10}} \).

Step 2: Apply the Power of a Power Property

For an expression \( (x^m)^{n} \), the exponents can be multiplied to simplify: \( x^{m \cdot n} \). Applying this to the expression \( (a^5 b^5)^{\frac{1}{10}} \) gives us \( a^{5 \cdot \frac{1}{10}} b^{5 \cdot \frac{1}{10}} \).

Step 3: Simplify Each Exponent

Perform the multiplication for each part of the expression. For \( a \), \( 5 \cdot \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \). Likewise, for \( b \), \( 5 \cdot \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \).

Step 4: Combine the Simplified Expression

Combine the simplified exponents to form the expression \( a^{\frac{1}{2}} b^{\frac{1}{2}} \). This is the expression simplified using rational exponents.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing them in their simplest form. A radical expression may look complicated, but there are techniques to unravel its complexity. By using rational exponents, we can transform these expressions into something more manageable. For instance, the radical \( \sqrt[10]{a^5 b^5} \) can be expressed as an exponential expression \( (a^5 b^5)^{\frac{1}{10}} \). Here’s how we can break it down:

• The nth root of a term is equivalent to raising that term to the power of \( \frac{1}{n} \).
• This transformation allows us to deal with exponents directly, instead of handling the root and the base values separately.

By transforming the radical into an exponential form, we ease the simplification process. Mastering this conversion process is key in simplifying complex radical expressions further using properties of exponents.

Exponential Expressions

An exponential expression is any mathematical expression that involves an exponent, which represents repeated multiplication. For example, \( x^3 \) indicates that \( x \) is multiplied by itself three times. When dealing with expressions such as \( (a^5 b^5)^{\frac{1}{10}} \), each component can be carefully simplified. Breaking down complex expressions forms a major part of dealing with exponentials. Here’s how:

• Each term inside the parentheses is handled individually.
• Apply the operation \( n^{\frac{1}{10}} \) to both \( a^5 \) and \( b^5 \), converting the expression into individual simpler term operations.

This method makes it easier to simplify each component, ensuring a clear pathway to solving exponential expressions effectively.

Power of a Power Property

The power of a power property is a fundamental rule while handling exponential expressions. It states that when raising an exponential expression to another power, you multiply the exponents. For a problem like \( (a^5 b^5)^{\frac{1}{10}} \), applying this rule will greatly simplify your work:

• The expression can be split into \( a^{5 \cdot \frac{1}{10}} \) and \( b^{5 \cdot \frac{1}{10}} \).
• Multiply the powers: \( 5 \cdot \frac{1}{10} = \frac{5}{10} \). This step simplifies both terms to \( a^{\frac{1}{2}} \) and \( b^{\frac{1}{2}} \).

Ultimately, using the power of a power property results in a neat and well-simplified expression. This highlights the advantage of understanding and applying exponent rules, substantially streamlining the simplification process.