Problem 41 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{m^{2} n^{7} p^{8}}$$

Step-by-Step Solution

Answer

\( p^2 \cdot \sqrt{m} \cdot n^{3/4} \cdot \sqrt[4]{n} \)

Step 1: Identify the Expression

The expression given is \( \sqrt[4]{m^{2} n^{7} p^{8}} \). This is a fourth-root expression involving variables \(m\), \(n\), and \(p\). Our goal is to simplify it.

Step 2: Apply the Property of Radicals

Recall the property that \( \sqrt[n]{a^b} = a^{b/n} \). This means that we can rewrite each term under the radical as a fractional exponent. Thus, \( m^2 \) becomes \( m^{2/4} \), \( n^7 \) becomes \( n^{7/4} \), and \( p^8 \) becomes \( p^{8/4} \).

Step 3: Simplify Each Exponent

Simplify each fraction in the exponents: \( m^{2/4} = m^{1/2} \) (since \(2/4 = 1/2\)), \( n^{7/4} \) remains as is (no simplification), and \( p^{8/4} = p^2 \) (since \(8/4 = 2\)).

Step 4: Express as a Single Radical

Combine the simplified expressions back under a single radical: \( \sqrt[4]{m^2 n^7 p^8} = (m^{1/2} n^{7/4} p^2)\). We can express this as \( p^2 m^{1/2} ^{7/4}\).

Step 5: Simplify Using Radical Notation

\( m^{1/2} \) can be expressed as \( \sqrt{m} \) and \( n^{7/4} \) means \( n^{3/4} n^{1/4} \), where \( n^{1/4} = \sqrt[4]{n}\). The final simplified radical expression is \( p^2 \cdot \sqrt{m} \cdot n^{3/4} \cdot \sqrt[4]{n} \).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that contain a radical sign, \(\sqrt{ }\), which indicates a root. They can include square roots, cube roots, or even higher-order roots like the fourth root. Recognizing these radicals is the first step towards simplifying them.

To simplify radical expressions, you need to understand the basic components:

• The radicand, which is the number or expression inside the radical.
• The index, which indicates the degree of the root. For example, a square root has an index of 2, while a fourth root has an index of 4.

By converting radical expressions into their equivalent forms using fractional exponents, it becomes easier to manipulate and simplify them further.

Properties of Radicals

Understanding the properties of radicals is crucial for simplifying complex expressions. These properties allow you to manipulate and break down radicals effectively. Some key properties include:

• Product Property: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\). This property allows you to combine multiplied radicals under a single radical.
• Quotient Property: \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}\). Use this property to divide radicals.
• Power Property: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). This property lets you express radicals using fractional exponents, which can simplify calculations greatly.

In our original exercise, the power property of radicals is especially useful. It allows each variable under the radical to be expressed in terms of a fractional exponent, leading to easier simplification.

Fractional Exponents

Fractional exponents provide a different perspective on handling radicals. They are an alternative way to write radical expressions that emphasize the relationship between roots and powers. For example, the expression \(\sqrt[n]{a}\) can be rewritten as \(a^{\frac{1}{n}}\).

Some steps on using fractional exponents include:

• Convert the radical to a fractional exponent for simplification.
• Simplify the fractional exponent by finding common factors.
• Convert back to radical form if needed, for the final expression.

In the given example, \(m^2\) was converted to \(m^{\frac{1}{2}}\), \(n^7\) remained as \(n^{\frac{7}{4}}\), and \(p^8\) was simplified to \(p^2\). These fractional exponents helped transform the original radical expression into a more manageable form, making it easier to interpret and simplify.