Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(x^{2} y^{13} \div y^{0}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{4}y^{26}\).

Step 1: Simplify Division Inside the Parentheses

First, recall that any non-zero number raised to the power of zero is 1. Thus, \(y^{0} = 1\). This means that dividing by \(y^{0}\) doesn't change the expression. Therefore, we can rewrite the expression as \((x^{2}y^{13})^{2}\).

Step 2: Apply the Power Rule

The power rule states that \((a^{m}b^{n})^{p} = a^{mp}b^{np}\). We apply this rule to the expression \((x^{2}y^{13})^{2}\) by multiplying the exponents inside the parentheses by the power outside. This gives us \(x^{2 \times 2}y^{13 \times 2}\), which simplifies to \(x^{4}y^{26}\).

Step 3: Write the Final Expression with Positive Exponents

Since our expression \(x^{4}y^{26}\) already contains positive exponents, we can conclude that this is the final simplified form of the given expression.

Key Concepts

Positive Exponents

Exponents are used to express repeated multiplication of a number by itself. A positive exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression \(x^{3}\), the base \(x\) is multiplied by itself three times: \(x \times x \times x\).

• Positive exponents describe quantities in multiplication.
• They are straightforward and maintain the original value of any number raised to a power.
• Always ensure conversions or simplifications leave exponents in their positive form to adhere to standard mathematical practices.

Understanding and working with positive exponents is crucial for mastering algebraic expressions. They help in organizing and simplifying calculations especially when dealing with polynomials. When an exercise asks to "write answers with positive exponents," it is a reminder to ensure that all exponents are displayed as such, never as negative unless specified by context.

Power Rule

The power rule in algebra is an important principle that simplifies expressions involving exponents. It states that when you raise an exponentiated base to another power, you can multiply the exponents. This is expressed as:

• \((a^{m})^{n} = a^{m \times n}\)
• applies to each term inside parentheses.
• helps condense expressions by reducing layers of exponents.

Applying the power rule helps to streamline complex problems by directly calculating new powers. For example, in the exercise, the expression \((x^{2}y^{13})^{2}\) used this rule, allowing us to multiply the exponents: \(x^{2 \times 2}\) and \(y^{13 \times 2}\), resulting in \(x^{4}y^{26}\).
This rule is essential in algebra because it simplifies the process of handling powers within powers, making calculations more efficient. It's especially useful in contexts where multiple layers of exponentiation appear.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves condensing an expression to its simplest form. This skill is about making expressions more manageable and easier to understand or compute. Simplifying might involve:

• Removing parentheses through distributive application of laws like the power rule.
• Combining like terms.
• Rewriting terms with a consistent format, such as using positive exponents.

In the step-by-step solution, the expression \(\left(x^{2} y^{13} \div y^{0}\right)^{2}\) was simplified first by recognizing \(y^{0}\) as 1, leaving the expression unchanged when divided. Next, the power rule simplified \((x^{2} y^{13})^{2}\) to \(x^{4}y^{26}\).
The goal of simplification is not only to reach a final, more condensed form but to do so logically and consistently. This practice makes solving equations and evaluating functions more efficient, especially important in complex algebraic contexts.