Problem 73 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[6]{m^{11}} $$

Step-by-Step Solution

Answer

\( m^{11/6} \)

Step 1: Understanding the Index of the Root

The given expression is \( \sqrt[6]{m^{11}} \). The number 6 here is the index of the root, which means we are looking for something raised to the 6th power that equals \( m^{11} \).

Step 2: Convert Radical to Exponent Form

Rewrite the expression \( \sqrt[6]{m^{11}} \) in an exponential form using the property \( \sqrt[n]{a} = a^{1/n} \). Therefore, \( \sqrt[6]{m^{11}} = (m^{11})^{1/6} \).

Step 3: Applying the Power of a Power Property

Use the exponent rule \( (a^m)^n = a^{m \cdot n} \) to simplify \( (m^{11})^{1/6} \). Multiply the exponents: \( 11 \cdot \frac{1}{6} \).

Step 4: Calculate the Exponent

Calculate \( 11 \cdot \frac{1}{6} = \frac{11}{6} \). Thus, the expression becomes \( m^{11/6} \).

Step 5: Final Simplification and Answer

Check if the fraction \( \frac{11}{6} \) can be simplified. Since it cannot, \( m^{11/6} \) is the simplest form of the expression.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and beyond. When you see a symbol like \( \sqrt[n]{a} \), it's called a radical, and the number \( n \) is the index of the root. In the exercise \( \sqrt[6]{m^{11}} \), the index 6 tells us that we're dealing with a sixth root.
When dealing with radical expressions, it's important to understand that you're looking for a number which, when raised to the power of the index, will give you the radicand (the number inside the root).
Having a strong grasp on this concept helps in recognizing potential simplifications when manipulating expressions.

Exponent Rules

Exponent rules are essential tools in simplifying expressions, especially when dealing with radicals and exponentiation. These rules give a structured way to handle powers and simplify expressions efficiently.

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

In the context of radical expressions like \( \sqrt[6]{m^{11}} \), converting to an exponent allows us to apply these rules directly. Transforming the radical to \( (m^{11})^{1/6} \) enables the usage of the power of a power rule, greatly simplifying the process.

Power of a Power Property

The power of a power property states that when you raise a power to another power, you multiply the exponents. In mathematical terms, \( (a^m)^n = a^{m \cdot n} \). This property is extremely useful in reducing complex expressions.
Using this property in the exercise \( (m^{11})^{1/6} \), we multiply the exponents: \( 11 \cdot \frac{1}{6} = \frac{11}{6} \). This simplifies the original expression to \( m^{11/6} \). By performing this reduction, we've transformed the problem from a radical expression to a simpler exponential form, making it easier to work with further if needed.