Problem 65 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{a^{16} b^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^4 b \).

Step 1: Recognize the Radical Expression

We are given the radical expression \( \sqrt[4]{a^{16} b^{4}} \). This is a fourth root, as indicated by the small 4 outside the radical symbol. We will simplify it by removing the radicals using the properties of exponents.

Step 2: Apply the Rule for Simplifying Radicals

The rule for simplifying a fourth root is to rewrite it as an equivalent expression using exponents: \( \sqrt[4]{x^n} = x^{n/4} \). We will apply this rule separately to each part of the expression inside the radical.

Step 3: Simplify \(a^{16}\)

Using the rule \( \sqrt[4]{x^n} = x^{n/4} \), simplify \( \sqrt[4]{a^{16}} \). This becomes \( a^{16/4} = a^4 \).

Step 4: Simplify \(b^4\)

Similarly, simplify \( \sqrt[4]{b^4} \) using \( \sqrt[4]{x^n} = x^{n/4} \). This results in \( b^{4/4} = b^1 = b \).

Step 5: Combine the Results

Now that each part of the radical has been simplified, combine them to get the final answer: \( a^4 \cdot b \).

Key Concepts

Properties of Exponents

Exponents are powerful tools that simplify expressions involving repeated multiplication. When dealing with radicals, a strong grasp of exponent properties can make simplification a breeze. The fundamental properties to remember include:

• Product of Powers: When multiplying same bases, add their exponents: \( x^a \times x^b = x^{a+b} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a \, b}\).
• Power of a Product: When raising a product to a power, distribute the exponent: \((xy)^a = x^a y^a\).
• Zero Exponent: Any nonzero number raised to the zero power is one: \( x^0 = 1\).

For radical expressions, especially fourth roots, using the property \( \sqrt[n]{x^m} = x^{m/n} \) can transform the problem into a form where these exponent rules are applied efficiently. Being comfortable with these properties will speed up solving and simplifying radical expressions significantly.

Fourth Root

The fourth root of a number is that number which, when raised to the power of four, gives us the original number. This can be written using a radical symbol as \( \sqrt[4]{x} \) or using exponents as \( x^{1/4} \). Understanding and manipulating fourth roots helps us when simplifying expressions like \( \sqrt[4]{a^{16} b^4} \).To simplify a fourth root:

• Express each term inside the radical using exponents.
• Apply the rule: \( \sqrt[4]{x^n} = x^{n/4} \).

For instance, \( \sqrt[4]{a^{16}} \) becomes \( a^{16/4} = a^4 \). Likewise, \( \sqrt[4]{b^4} \) simplifies to \( b^{4/4} = b \). Thus, fourth roots effectively break down complex radicals into simple and manageable terms.

Radical Expressions

Radical expressions include symbols such as the square root, cube root, or in this case, the fourth root. They involve roots of a number or an algebraic expression. Simplifying radical expressions is crucial for all levels of algebra, as it makes computations more manageable and expressions easier to interpret.To simplify radical expressions:

• Identify the type of root involved (e.g., square root, fourth root).
• Use familiar roots like \( \sqrt{x} = x^{1/2} \) or \( \sqrt[4]{x} = x^{1/4} \).
• Convert complex roots using the proper exponent rules (as outlined in properties of exponents).

For example, converting \( \sqrt[4]{a^{16} b^4} \) into exponents simplifies it to \( a^4 \cdot b \), a much clearer expression. Understanding radical expressions in this way aids in solving algebra problems more effectively.

Positive Real Numbers

When simplifying radicals, assuming that all variables represent positive real numbers is important. This assumption ensures that the expressions remain real and avoids complex numbers, which arise with negative numbers under even roots.Positive real numbers possess properties like:

• They are greater than zero and can be found along the number line anywhere to the right of zero.
• Operations involving them (like multiplication, division, or taking roots) will always yield positive real results.
• They simplify the square roots, or any even roots, because the operation is valid only for non-negative inputs.

This assumption simplifies scenarios such as \( \sqrt[4]{a^{16} b^4} \), as you do not have to consider imaginary or complex solutions. Working with positive real numbers allows students to focus on the core concepts without extra complexities.