Problem 17 - Beginning and Intermediate Algebra

Question

Simplify. $$(x+3)^{4}(x+3)^{-2}$$

Step-by-Step Solution

Answer

The simplified expression for \((x+3)^4 (x+3)^{-2}\) is \((x+3)^2\).

Step 1: Apply the Product of Powers Property

Recall that the product of powers property states that \(a^m a^n = a^{m+n}\) for any base a and integers m and n. Here, our base is (x + 3). Thus, we can apply the property to our expression: \((x+3)^4 (x+3)^{-2} = (x+3)^{4 + (-2)}\)

Step 2: Combine Exponents

Now, we combine the exponents by performing the addition operation: \((x+3)^{4 + (-2)} = (x+3)^{2}\)

Step 3: Final Answer

The simplified expression is: \((x+3)^2\)

Key Concepts

Simplifying Expressions

Simplifying expressions means to reduce them to their simplest form. This makes complex math problems easier to handle. In algebra, simplification often involves combining like terms, reducing fractions, or condensing powers and roots.
For example, in the expression \((x+3)^4 (x+3)^{-2}\), the goal is to simplify it down to the most compact form possible without changing its value.
To simplify such expressions, rules about exponents and algebra must be applied carefully. Simplifying involves handling the terms logically, step by step, until you reach a point where no further simplification is possible. It's like cleaning up a messy room; you put everything in its right place to make the room look neat.

Product of Powers Property

The Product of Powers Property is a fundamental rule when working with exponents. It simplifies expressions where the same base is multiplied by itself several times, each with different exponents. The property tells us that when we multiply powers with the same base, we can simply add the exponents.

• For example, the property is written as \(a^m a^n = a^{m+n}\). Here, \(a\) is the base, and \(m\) and \(n\) are the exponents.
• This rule significantly helps in simplifying multiplication of expressions like \((x+3)^4 (x+3)^{-2}\).
• By adding the exponents \(4\) and \(-2\), we can write \((x+3)^{4+(-2)}\), which simplifies to \((x+3)^2\).

Understanding the Product of Powers Property can make solving exponential expressions much simpler and faster.

Combining Exponents

Combining exponents is a critical step in simplifying expressions involving powers. When you have expressions with the same base in a multiplication, combining exponents simplifies the expression into a single term with just one exponent.
In the expression \((x+3)^{4+(-2)}\), you combine the exponents directly by performing addition or subtraction depending on their signs.

• Addition if both are positive: \(a^{3} \times a^{2} = a^{3+2} = a^{5}\).
• Subtraction if one is negative: \(a^{4} \times a^{-2} = a^{4-2} = a^{2}\).

The process of combining exponents streamlines the computation and helps in reaching the most simplified form of the expression, which for our example is \((x+3)^2\). This final form represents the expression in a concise and easily manageable way.