Problem 76 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 3 \sqrt[4]{x^{4} y}-2 \sqrt[4]{x^{4} y} $$

Step-by-Step Solution

Answer

The simplified expression is \(x \cdot \sqrt[4]{y}\).

Step 1: Identify Like Terms

First, notice that both terms in the expression \(3 \sqrt[4]{x^{4} y} - 2 \sqrt[4]{x^{4} y}\) have the same radical part: \(\sqrt[4]{x^{4} y}\). This means they are like terms and can be combined.

Step 2: Factor Out the Common Term

Because both terms have \(\sqrt[4]{x^{4} y}\), we can factor this out: \[ 3 \sqrt[4]{x^{4} y} - 2 \sqrt[4]{x^{4} y} = (3 - 2) \sqrt[4]{x^{4} y}. \]Notice that we took the coefficient of each term (3 and 2) and subtracted them.

Step 3: Simplify the Coefficients

Simplifying the coefficients gives us:\[ (3 - 2) \sqrt[4]{x^{4} y} = 1 \cdot \sqrt[4]{x^{4} y}. \] This simplifies to: \[ \sqrt[4]{x^{4} y}. \]

Step 4: Simplify the Radical Expression

Since \(x^{4} y\) is inside the fourth root and the exponent of \(x\) is 4, which matches the radical index, \(x\) can be taken out of the radical completely. Therefore, \[ \sqrt[4]{x^{4} y} = x \cdot \sqrt[4]{y}. \]

Key Concepts

Like Terms

In algebra, like terms are terms that contain the same variables raised to the same power. This means they are the parts of an expression that are similar and can be added or subtracted directly. Consider the expression from the exercise:

• Like terms: Both terms, \(3 \sqrt[4]{x^{4} y}\) and \(2 \sqrt[4]{x^{4} y}\), have the identical radical part \(\sqrt[4]{x^{4} y}\).
• Thus, they can be combined by directly dealing with their coefficients, ignoring the radical part temporarily when adding or subtracting.

Recognizing like terms simplifies algebraic expressions by allowing straightforward arithmetic on the coefficients. This direct handling of coefficients significantly reduces the complexity of an expression.

Fourth Root

The fourth root of a number, expressed as \(\sqrt[4]{x}\), is a special type of radical expression. It represents the value which, when multiplied by itself four times, equals the original number. Breaking down the expression step-by-step:

• For example: \(\sqrt[4]{x^4 y}\) effectively asks what number raised to the fourth power results in \(x^4 y\)?.

• Role in Simplification

  In our exercise, applying the fourth root allows us to simplify terms such as \(x^4\), bringing \(x\) outside of the radical completely.

This simplification aspect is crucial in reducing radical complexities in expressions and algebraic equations.

Coefficients

Coefficients are the numerical parts of terms that include variables. When dealing with expressions containing like terms, coefficients are crucial:

• The expression \(3 \sqrt[4]{x^{4} y} - 2 \sqrt[4]{x^{4} y}\) has coefficients 3 and 2 for the like radical term \(\sqrt[4]{x^{4} y}\).

• Combining Coefficients

  Combine coefficients of like terms by standard arithmetic operations, here by subtraction: \(3 - 2 = 1\).
• The resulting expression becomes \(1 \cdot \sqrt[4]{x^{4} y}\), making the expression less cluttered.

By understanding how to manipulate coefficients, simplifying expressions becomes a much more manageable task.

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. It can involve several steps:

• The original radical expression \(\sqrt[4]{x^{4} y}\) can often be simplified by taking terms out of the radical.

• Process and Strategy

  In this specific exercise, since \(x^{4}\) matches the radical index, \(x\) is taken out of the radical: \(\sqrt[4]{x^{4} y} = x \cdot \sqrt[4]{y}\).
• This process leverages the property that something raised to the power of the same number as the root can be extracted from the radical.

Simplifying radicals makes it easier to work with expressions and find further arithmetic or algebraic solutions.