Problem 119 - Intermediate Algebra

Question

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[4]{a^{12} b^{4} c^{20}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^3 b c^5 \).

Step 1: Recognize the Expression Format

The expression is a fourth root, expressed as \( \sqrt[4]{a^{12} b^{4} c^{20}} \). The objective is to simplify this root using the properties of exponents.

Step 2: Apply the Root to Each Factor

When you take the fourth root of a product, you can apply the root to each term separately. The expression becomes \( \sqrt[4]{a^{12}} \times \sqrt[4]{b^{4}} \times \sqrt[4]{c^{20}} \).

Step 3: Simplify \( \sqrt[4]{a^{12}} \)

For \( \sqrt[4]{a^{12}} \), use the property that \( \sqrt[n]{x^m} = x^{m/n} \). Here, \( a^{12/4} = a^3 \).

Step 4: Simplify \( \sqrt[4]{b^4} \)

Similar for \( \sqrt[4]{b^4} \), this is \( b^{4/4} = b^1 = b \).

Step 5: Simplify \( \sqrt[4]{c^{20}} \)

For \( \sqrt[4]{c^{20}} \), this becomes \( c^{20/4} = c^5 \).

Step 6: Combine the Simplified Terms

Now, combine the results from the previous steps: \( a^3 \times b \times c^5 \). Therefore, the simplified expression is \( a^3 b c^5 \).

Key Concepts

Understanding Roots in Algebra

Roots represent the inverse operation of exponents. When we talk about a fourth root, as in the expression \( \sqrt[4]{a^{12} b^{4} c^{20}} \), it means we want to find a value that when raised to the power of four gives us the original expression back.

Think of it as the opposite of exponentiation. If you have \( x^n \), then \( \sqrt[n]{x} \) gives you the base of the exponentiation.

For example, \( \sqrt[4]{16} \) is 2 because \( 2^4 = 16 \).

• The operation can be applied separately to each factor in a product, making it easier to simplify complex expressions.
• Using the property \( \sqrt[n]{x^m} = x^{m/n} \) allows us to transform roots into exponents, greatly facilitating simplification.

This understanding helps in breaking down the expression into manageable parts, simplifying the work required.

Simplification Using Exponents

Simplification involves reducing expressions to their simplest form. When dealing with expressions like \( \sqrt[4]{a^{12} b^{4} c^{20}} \), we use the relationship between roots and exponents.

By converting roots into exponent expressions, we can handle each term separately:

\( \sqrt[4]{a^{12}} \) becomes \( a^{12/4} = a^3 \).

Similarly, \( \sqrt[4]{b^4} \) transitions to \( b^{4/4} = b \) and \( \sqrt[4]{c^{20}} \) becomes \( c^{20/4} = c^5 \).

• You separate the expression by treating each variable independently.
• This method helps manage the power of variables distinctively.

Eventually, the goal is to simplify the expression into a form like one of these where no roots or unnecessary powers remain, just the product of variables raised to powers.

Algebraic Expressions and Their Components

Algebraic expressions combine numbers and variables using operations such as addition, subtraction, multiplication, or division.

Variables can have powers, and simplifying these expressions involves understanding how to manipulate these powers.

For instance, in the expression \( a^3 b c^5 \), this is a simplified version of the more complex original involving roots and powers.

• Each component of the expression, such as \(a^3\), represents a term.
• Knowing the law of exponents is crucial, such as \( (x^a)(x^b) = x^{a+b} \) or \( (x^a)/(x^b) = x^{a-b} \).

Understanding and simplifying algebraic expressions involves breaking down and recomposing the elements using established mathematical rules. This simplification makes complex expressions easier to handle and further solve equations or evaluate.