Problem 42 - College Algebra

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: Understand the Expression

The given expression is \(\left(x^{2} y^{13} \div y^{0}\right)^{2}\). We need to simplify this expression step by step, ensuring the final answer contains positive exponents.

Step 2: Simplify the Division Inside the Parentheses

Start by simplifying \(x^{2} y^{13} \div y^{0}\). Note that dividing by \(y^{0}\) effectively means multiplying by 1, since \(y^{0} = 1\). Therefore, \(x^{2} y^{13} \div y^{0} = x^{2} y^{13}\).

Step 3: Apply the Power of a Product Property

The expression now is \((x^{2} y^{13})^{2}\). Use the power of a product rule, which states \((a^m b^n)^p = a^{m\cdot p} b^{n\cdot p}\). This turns into \((x^2)^{2} \cdot (y^{13})^{2}\).

Step 4: Calculate the Exponents

Apply the power to each term: \((x^2)^2 = x^{2 \times 2} = x^4\) and \((y^{13})^2 = y^{13 \times 2} = y^{26}\).

Step 5: Combine the Results

The simplified expression is \(x^4 y^{26}\). This expression has only positive exponents, which meets the requirement of the problem.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. If you see a number with a tiny number above and to the right of it, that's an exponent indicating how many times you multiply the base by itself. For example, in the term \(x^2\), \(x\) is the base and 2 is the exponent, meaning \(x\) multiplied by itself, or \(x \times x\).
Using exponents makes it easier to write and calculate with large or repeated numbers.

• Exponents simplify expressions by indicating repeated multiplication.
• They allow for easier computation and manipulation of numbers.
• Understanding exponents is crucial in algebra, as it helps in simplifying complex expressions.

When working with exponents, remember that any number to the zero power is 1. This principle is useful when simplifying expressions like dividing by \(y^0\), which simply becomes 1.

Power of a Product Property

The power of a product property is a key concept in simplifying expressions with exponents. This rule states that when you have an expression inside parentheses raised to an exponent, you can distribute the exponent to each factor inside the parentheses. In formula terms, \[(a^m b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\]. This property helps in breaking down more complex expressions into simpler ones.
In our expression example, \[(x^2 y^{13})^2\], instead of dealing with it as a whole, you apply the power separately to each component:

• Apply the power to \(x^2\) resulting in \(x^{2\cdot2} = x^4\).
• Apply the power to \(y^{13}\) resulting in \(y^{13\cdot2} = y^{26}\).

Using this property correctly will simplify problems and lead to expressions with positive exponents, making it easier to work with them in algebraic operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition and multiplication) put together to represent a value. These expressions can include exponents, which tell how many times a number is used in a multiplication.
An example is \(x^2 y^{13}\), where \(x\) and \(y\) are variables. You may encounter expressions requiring simplification by applying certain rules and properties. This is where understanding exponents and properties like the power of a product becomes valuable.
Algebraic expressions:

• Help in forming equations to represent real-world situations.
• Can be simplified by understanding certain mathematical properties.
• Require careful manipulation to maintain equality when simplified.

In the context of simplifying expressions, combining like terms, applying exponent rules correctly, and using properties such as the power of a product can change a complex-looking expression into a straightforward form, as seen in the final result \(x^4 y^{26}\).