Problem 456 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[4]{m^{8}}\) (b) \(\sqrt[5]{n^{20}}\)

Step-by-Step Solution

Answer

(a) \( m^{2} \) (b) \( n^{4} \)

Step 1: Understand the Radical and Exponent Properties

Understand that a root can be rewritten as a fractional exponent. Specifically, \( \sqrt[b]{a^c} = a^{\frac{c}{b}} \ \). This will be used to simplify the expressions.

Step 2: Simplify Part (a) \( \sqrt[4]{m^8} \)

Rewrite \( \sqrt[4]{m^8} \) as an exponent: \( (m^8)^{\frac{1}{4}} \). Then simplify the exponent by multiplying: \( m^{8 \cdot \frac{1}{4}} = m^{2} \). So, \( \sqrt[4]{m^8} = m^2 \).

Step 3: Simplify Part (b) \( \sqrt[5]{n^{20}} \)

Rewrite \( \sqrt[5]{n^{20}} \) as an exponent: \( (n^{20})^{\frac{1}{5}} \). Simplify the exponent by multiplying: \( n^{20 \cdot \frac{1}{5}} = n^{4} \). Therefore, \( \sqrt[5]{n^{20}} = n^4 \).

Key Concepts

Fractional Exponents

Fractional exponents might seem confusing at first, but they are quite handy once you understand them. A fractional exponent is another way to represent a root. For instance, the square root of a number can be written as that number raised to the power of one-half, or \(a^{1/2}\). In general, the b-th root of a number raised to the c power can be written as \(a^{c/b}\).
Here is how it works:

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Radical Expressions

A radical expression involves roots, such as square roots, cube roots, etc. The notation \(\sqrt[b]{a}\) represents the b-th root of a. Radicals can often be simplified by converting them to fractional exponents. This allows you to use the rules of exponents to simplify the expression further.
Let's break it down:

• Consider the expression \(\sqrt[b]{a^c}\). Use the property that \(\sqrt[b]{a^c} = a^{\frac{c}{b}}\).
  
• For example, \(\sqrt[4]{m^8}\) becomes \(m^{8/4}\), which simplifies to \(m^2\).
  
• Similarly, \(\sqrt[5]{n^{20}}\) becomes \(n^{20/5}\), which simplifies to \(n^4\).

Algebraic Simplification

Simplifying algebraic expressions often means reducing them to their simplest form. This can involve combining like terms, reducing fractions, or using exponent rules.
When dealing with fractional exponents or radicals, the goal is to convert the expression into a simpler form.

• Starting with the given radical, rewrite it using a fractional exponent.
  
• Next, multiply the exponents according to the rules of exponents.
  
• Finally, simplify the resulting expression.
  
  For example:
  
  \(\sqrt[4]{m^8}\) becomes \( (m^8)^{1/4} = m^{8/4} = m^2\).
  
  Similarly, \(\sqrt[5]{n^{20}}\) becomes \( (n^{20})^{1/5} = n^{20/5} = n^4\).
  
  Understanding these steps will help you simplify many algebraic expressions with ease!