Problem 83 - Precalculus: Functions and Graphs

Question

Simplify the expression, assuming \(x\) and \(y\) may be negative. $$\sqrt[4]{x^{8}(y-1)^{12}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(x^2(y-1)^3\).

Step 1: Rewrite the Expression with Rational Exponents

The expression is \( \sqrt[4]{x^{8}(y-1)^{12}} \). We first rewrite the fourth root as a power of 1/4: \[\sqrt[4]{x^{8}(y-1)^{12}} = (x^{8}(y-1)^{12})^{1/4}.\] Then, distribute the exponent of \(1/4\) to each factor inside the parentheses:\[(x^{8})^{1/4} \cdot ((y-1)^{12})^{1/4}.\]

Step 2: Simplify Individual Terms

Apply the power rule of exponents, \((a^m)^n = a^{m \cdot n}\), to simplify each term separately:- For \((x^{8})^{1/4}\), we have \(x^{8 \cdot \frac{1}{4}} = x^{2}\).- For \(((y-1)^{12})^{1/4}\), we have \((y-1)^{12 \cdot \frac{1}{4}} = (y-1)^{3}\).

Step 3: Combine the Simplified Terms

Multiply the results from Step 2 to get the simplified expression:\[x^{2} \cdot (y-1)^{3}.\] This is the simplified form of the given expression.

Key Concepts

Rational Exponents

Rational exponents provide a convenient way to express roots as powers. Rather than writing a root symbol, we use a fraction as the exponent. For example, the fourth root of a number can be expressed with an exponent of 1/4. This means:

• The fourth root of a number \( a \) is \( a^{1/4} \).
• This approach is especially helpful when dealing with variables and powers.

Using rational exponents helps in algebraic manipulation since they follow the same rules as whole number exponents. In our example, we used the expression \((x^8 (y-1)^{12})^{1/4}\) to signify that we are taking the fourth root of the entire expression.

Power Rule

The power rule is a fundamental principle in dealing with exponents. It states:
\((a^m)^n = a^{m \cdot n}\).
This allows us to simplify expressions where an exponent is raised to another exponent.

• In our expression, \((x^8)^{1/4}\), the power rule helps simplify it to \(x^{8 \cdot 1/4} = x^2\).
• Similarly, for \(((y-1)^{12})^{1/4}\), applying the power rule gives \((y-1)^{12 \cdot 1/4} = (y-1)^{3}\).

This approach breaks down the problem, making it manageable and straightforward to simplify complex expressions.

Simplifying Expressions

Simplifying expressions usually means reducing them to the simplest possible form. This might involve combining like terms, factoring, or applying exponent rules.

• In the example, we began with a fourth root and used rational exponents to rewrite the expression.
• We applied the power rule to handle each term individually.
• After simplification, the expression \(x^{2} \cdot (y-1)^{3}\) no longer contained roots and was expressed in its simplest polynomial form.

These steps ensure clarity and make the expression much easier to interpret and use in further mathematical operations.

Fourth Root

The fourth root of a number is the value that, when multiplied by itself four times, gives the original number. It is written as \( \sqrt[4]{a} \), which can also be expressed as \( a^{1/4} \).

• Finding the fourth root is essentially finding \( a \) such that \( a^4 = x \).
• When dealing with variables, expressing the fourth root as an exponent makes the expression easier to simplify.
• In the exercise \( \sqrt[4]{x^{8}(y-1)^{12}} \), the fourth root is calculated across the entire expression, allowing use of exponent rules for simplification.

Understanding fourth roots in this way integrates well with other algebraic operations, like using rational exponents and applying power rules.