Problem 24 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ x y^{-3} $$

Step-by-Step Solution

Answer

The simplified form of \(x y^{-3}\) is \(x/y^3\).

Step 1: Understanding the Exponential Expression

The given expression is \(x y^{-3}\). Here, \(y^{-3}\) has a negative exponent, which means it is the reciprocal of \(y^3\). So it can be written as \(1/y^3\).

Step 2: Applying the Negative Exponent Rule

The expression \(x y^{-3}\) can be rewritten as \(x * 1/y^3\) according to the rule of negative exponents. With multiplication, we don't need to explicitly write the multiplication symbol. This simplifies to \(x/y^3\).

Step 3: Final Answer

Therefore, the expression \(x y^{-3}\) simplifies to \(x/y^3\). This is the simplified form of the given expression.

Key Concepts

Understanding Exponential Expressions

Exponential expressions are powerful mathematical tools used to express repeated multiplication of a base number. In the expression like \(x y^{-3}\), the base is \(y\) and the exponent is \(-3\). Here is what that means:

• The base denotes the number or variable being multiplied.
• The exponent indicates how many times to use the base in a multiplication.

When dealing with negative exponents, it's important to understand that they represent the concept of taking a reciprocal. This shifts our view from repeated multiplication to repeated division, which directly links to our next topic: reciprocals.

Exploring Reciprocal Concepts

The reciprocal of a number or expression is simply one divided by that number or expression. For a given number \(a\), the reciprocal is \(1/a\). When it comes to exponential expressions with negative exponents, such as \(y^{-3}\), the negative sign indicates that we are dealing with a reciprocal. Thus, \(y^{-3}\) becomes \(1/y^3\).

• The negative exponent tells us to "flip" the base, creating a fraction where the base with a positive exponent of the same value is the denominator.
• This particular rule allows you to effortlessly transform expressions and simplify complex equations.

Grasping the concept of reciprocals is crucial for simplifying expressions involving negative exponents.

Steps in Simplification of Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining the same mathematical value. Let's break down the process using our example, \(x y^{-3}\):

• First, identify components with negative exponents - here, it's \(y^{-3}\).
• Convert negative exponents by writing them as reciprocals. So, \(y^{-3}\) becomes \(1/y^3\).
• Combine and simplify the expression. When you see \(x * 1/y^3\), it can be simplified to \(x/y^3\).

Using these steps, you not only simplify equations but also gain a deeper understanding of how different mathematical operations interact. This understanding is key to solving more complex mathematical problems.