Problem 102 - College Algebra Essentials

Question

In Exercises \(101-108\), simplify by reducing the index of the radical. $$\sqrt[4]{7^{2}}$$

Step-by-Step Solution

Answer

\(\sqrt{7}\)

Step 1: Analysis

The given expression is \(\sqrt[4]{7^{2}}\). Here, n (the root or index of the radical) is 4 and m (the exponent) is 2. Your task is to simplify it according to the root-power rule.

Step 2: Applying the root-power rule

According to the rule, \(\sqrt[n]{a^{m}} = a^{m/n}\). In our case, a is 7. Applying the rule to our expression gives us: \(7^{2/4} = 7^{0.5}\)

Step 3: Simplify

Finally, \(7^{0.5}\) is just the square root of 7, or \( \sqrt{7} \). This is our final simplified expression.

Key Concepts

Root-Power Rule

When dealing with radical expressions, one useful tool is the Root-Power Rule. It allows you to express radicals in terms of exponents, making them easier to manipulate and simplify. The rule states that the nth root of a number raised to the power of m can be written as the number raised to the power of m/n. In formula terms, it looks like this:

• \( \sqrt[n]{a^{m}} = a^{m/n} \)

To better understand this, let's apply the rule to our example, \( \sqrt[4]{7^{2}} \). Here, the root is 4, and the exponent is 2, so applying the Root-Power Rule transforms the expression into \( 7^{2/4} \). This means you are raising 7 to the power of 0.5 instead of taking a fourth root, which simplifies further calculations.

Simplifying Radicals

Radical simplification is a method used to rewrite complex radical expressions in a more workable form. By utilizing tools like the Root-Power Rule, you can transform the expression into one that is easier to interpret and use in further calculations.In our example given as \( \sqrt[4]{7^{2}} \), we use the Root-Power Rule to express it as \( 7^{0.5} \). This is a simpler representation because it converts the fourth root into a more familiar square root, \( \sqrt{7} \). Simplifying radicals ultimately helps in clearing up complicated mathematical expressions and can ease the process in algebraic operations later on.Remember:

• Express radicals with fractional exponents for ease.
• Always look for the simplest form of the expression.

Exponents

Exponents are a numerical shorthand that denote repeated multiplication of a number by itself. For instance, \( 7^{2} \) means 7 multiplied by 7. This concise form allows for easier manipulation in equations and expressions. In the context of radicals, exponents are critical because they allow for conversions between radical and exponential forms.Rewriting a radical using an exponent, as is done with the Root-Power Rule, aids in simplifying expressions. For the expression \( \sqrt[4]{7^{2}} \), writing it as \( 7^{0.5} \) allows us to use basic rules of exponents for further operations. Key points about exponents:

• \( a^{0.5} \) represents the square root of \( a \).
• Fractions as exponents indicate roots: a fraction like \( \frac{m}{n} \) suggests the nth root.
• Understanding how to convert between fractional exponents and radicals is essential for simplification.

Algebraic Simplification

Algebraic simplification is the process of making a mathematical expression as straightforward as possible. It involves reducing the number of operations or terms, making equations easier to handle.When it comes to simplifying the radical \( \sqrt[4]{7^{2}} \), transforming it into an exponential form \( 7^{0.5} \) and further reducing it to \( \sqrt{7} \) simplifies the expression. This not only aids in computations but also in recognizing patterns and relationships in math problems.The importance of algebraic simplification can be summarized:

• Allows for cleaner, easier-to-read expressions.
• Facilitates further algebraic operations and problem-solving.
• Promotes deeper understanding of mathematical structures and concepts.