Problem 27 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(432x^7\).

Step 1: Apply the Power of a Power Property

First, we need to apply the power of a power property of exponents. The power of a power property states that \( (a^m)^n = a^{(m \times n)} \). Using this, we can simplify the given expression: \((2x)^4 (3x)^3 = (2^4 x^4) (3^3 x^3)\)

Step 2: Calculate the numerical powers

Calculate the numerical powers: \(2^4 = 16\) and \(3^3 = 27\) Now the expression becomes: \((16x^4) (27x^3)\)

Step 3: Apply the Product of Powers Property

We need to apply the product of powers property of exponents. The product of powers property states that \( a^m \times a^n = a^{(m + n)} \). Using this, we can simplify the expression: \((16x^4) (27x^3) = 16 \times 27 x^{(4+3)}\)

Step 4: Simplify the coefficients and exponents

Now, we just need to multiply the coefficients and simplify the exponents: \(16 \times 27 x^{(4+3)} = 432x^7\) So, the simplified expression is: \[432x^7\]

Key Concepts

Power of a Power Property

The "Power of a Power Property" is a rule that makes simplifying expressions involving exponents much easier. When you have a number or a variable raised to a power, and that whole expression is raised to another power, this property comes into play. The rule states:

• \((a^m)^n = a^{(m \times n)}\)

Imagine you're stacking powers. Instead of multiplying your base number repeatedly, you simply multiply the powers together. This keeps things clean and ensures your computations are straightforward. For instance, applying this to the expression
\((2x)^4\), we look at it as \(2^4\) and \(x^4\), allowing us to treat each component separately. This results in \((2^4 \times x^4)\). Similarly, \((3x)^3\) becomes \((3^3 \times x^3)\). Understanding this property helps you quickly break down complicated exponential expressions.

Product of Powers Property

Another handy tool for working with exponents is the "Product of Powers Property." This property is essential when you're multiplying terms with the same base raised to different powers. It tells you that you can simply add the exponents while using the same base, like this:

• \(a^m \times a^n = a^{(m + n)}\).

The concept here is straightforward. It stems from the idea that multiplying is like repeated addition. Therefore, when you multiply similar bases, you count all those powers together. In our original problem, we used this principle on the \((x^4 \times x^3)\). This boils down to \(x^{4+3} = x^7\).
This property simplifies the work involved especially in lengthy calculations involving larger expressions with the same base.

Simplifying Expressions

"Simplifying Expressions" is an umbrella term for making complicated mathematical expressions more manageable. It involves multiple steps, including the use of properties such as power of a power and product of powers. One of the ultimate goals of simplification is to express calculations in their simplest form, making them easier to understand and work with.

• Start by breaking down the problem into smaller, manageable parts using known properties of exponents.
• Calculate any numerical powers, like \(2^4 = 16\) and \(3^3 = 27\) in our example.
• Combine terms using the properties (as we did with \(x^4 \times x^3 = x^7\)).
• Multiply coefficients together (for instance, \(16 \times 27 = 432\)).

This step-by-step process leads you to a final expression that is tidy, efficient, and often easier to interpret, much like our result of \(432x^7\). Learning to simplify effectively is a skill that's valuable in all levels of math. It makes your work cleaner and your solutions more accurate.