Problem 52 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt[4]{4 b c^{3}} \cdot \sqrt[4]{64 a b^{3} c^{2}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \( 4b \sqrt[4]{ac^5} \).

Step 1: Rewrite the Expression Individually

To simplify \( \sqrt[4]{4b c^3} \cdot \sqrt[4]{64 a b^3 c^2} \), start by rewriting each expression as a separate fourth root: \( \sqrt[4]{4b c^3} \) and \( \sqrt[4]{64a b^3 c^2} \).

Step 2: Break Down Inside the Roots

Express each term within the roots as a product of perfect fourth powers and remainders:\[ \sqrt[4]{4b c^3} = \sqrt[4]{4 \cdot b \cdot (c^3)} = \sqrt[4]{2^2 \cdot b \cdot (c^3)} \]\[ \sqrt[4]{64a b^3 c^2} = \sqrt[4]{64 \cdot a \cdot (b^3) \cdot (c^2)} = \sqrt[4]{2^6 \cdot a \cdot b^3 \cdot (c^2)} \]

Step 3: Simplify Each Root Separately

Now simplify each root by factoring out perfect fourth powers:\[ \sqrt[4]{2^2 \cdot b \cdot c^3} = \sqrt[4]{(2^2)} \cdot \sqrt[4]{b} \cdot \sqrt[4]{(c^3)} \]Since none are perfect fourth powers, this expression remains as is.\[ \sqrt[4]{2^6 \cdot a \cdot b^3 \cdot c^2} = \sqrt[4]{(2^4) \cdot (2^2) \cdot a \cdot b^3 \cdot c^2} = 2 \cdot \sqrt[4]{2^2 \cdot a \cdot b^3 \cdot c^2} \]

Step 4: Combine the Simplified Parts

Multiply the simplified forms of the radicals:\[ \sqrt[4]{4b c^3} \cdot \sqrt[4]{64a b^3 c^2} = \sqrt[4]{2^2b c^3} \cdot 2 \cdot \sqrt[4]{2^2ab^3c^2} \]Simplify further by combining under one radical:\[ 2 \cdot \sqrt[4]{(2^2b c^3)(2^2ab^3c^2)} \]

Step 5: Simplify Expression Inside the Combined Root

Simplify the expression under the root:\[ \sqrt[4]{(2^2 b c^3)(2^2 ab^3 c^2)} = \sqrt[4]{(2^4 a b^4 c^5)} \]Factor out the perfect fourth powers:\[ \sqrt[4]{(2^4)(b^4)} \cdot \sqrt[4]{(a c^5)} = 2b \cdot \sqrt[4]{a c^5} \]

Step 6: Final Simplified Expression

Combine the factors to get the final expression:\[ 2 \cdot 2b \cdot \sqrt[4]{a c^5} = 4b \cdot \sqrt[4]{a c^5} \]This is the simplified form of the original expression.

Key Concepts

Fourth Roots Made Easy

A fourth root is a number that, when raised to the power of four, gives the original number. Simply put, if you want to find the fourth root of a number, you're looking for a number that multiplied by itself three more times equals the original number. For example, 2 is the fourth root of 16 because \(2^4 = 16\).

When dealing with expressions like \( \sqrt[4]{4bc^3} \), you might encounter non-perfect roots. These can often be simplified by finding the largest factor that is a perfect fourth power within the number. This means breaking down the expression into manageable parts that can be individually simplified before recombining.

For variables and numbers inside such roots, you can break them down using prime factorization or simply evaluate whether the terms can be expressed as a product of integer powers that match the root. Understanding fourth roots helps in simplifying complex algebraic expressions.

Decoding Perfect nth Powers

Perfect nth powers are numbers that can be written as an integer raised to the power of n. In our case here, since we are dealing with fourth roots, the perfect fourth power involves finding a number where an integer raised to the fourth power equals the given number. Examples include numbers like 16 (since \(2^4 = 16\)) and 81 (since \(3^4 = 81\)).

When simplifying expressions, we aim to identify these perfect powers within the larger expression. In the context of the problem, the number 64 is a perfect fourth power since \(2^6\) can be broken down into \((2^4)\cdot(2^2)\), where \(2^4\) is a perfect fourth power.

For algebraic variables, we follow a similar principle. For instance, finding a perfect fourth power for \(b^3\) involves expressing it as \(b\cdot b^2\) without a perfect fourth power in \(b^3\). Identifying these forms helps in the simplification process.

Unlocking Factorization in Algebra

Factorization in Algebra is the process of breaking down numbers or expressions into multiples or factors that when multiplied, give back the original number or expression. In the case of simplifying expressions with roots, factorization allows us to identify parts of an expression that we can pull out or simplify further.

Consider the expression \( \sqrt[4]{2^6ab^3c^2} \). Using factorization, we can break it down as \( \sqrt[4]{(2^4)(2^2)ab^3c^2} \). This is done by identifying parts of the expression (like \(2^4\)) that are perfect fourth powers and can be taken out of the root.

For variables, if we have powers that coincide with the root, like \(b^4\), they can be isolated and factored out, simplifying the expression further. Factorization is a key tool and skill in algebra that often leads to simpler expressions and solutions, making it an essential part of your algebraic toolkit.