Problem 42 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(2 a^{3} a^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 16a^{20} \).

Step 1: Multiply the Exponents Inside the Parentheses

The given expression is \( (2a^3a^2)^4 \). First, simplify the expression inside the parentheses by multiplying the exponents of \( a \). Since \( a^3 \) and \( a^2 \) are both powers of \( a \), add their exponents together: \( a^{3+2} = a^5 \). The expression inside the parentheses becomes \( 2a^5 \).

Step 2: Apply the Power to Each Component Inside the Parentheses

Now, distribute the exponent \( 4 \) outside the parentheses to each term inside. For \( 2a^5 \), raise \( 2 \) to the power of \( 4 \) and \( a^5 \) to the power of \( 4 \). Calculate \( 2^4 = 16 \).

Step 3: Use the Power of a Power Property on the Variable

Apply the power rule \((a^m)^n = a^{m imes n}\) to the variable part \( a^5 \). Multiply the exponents: \( a^{5 imes 4} = a^{20} \).

Step 4: Write the Final Expression

Combine the results from steps 2 and 3 to form the final expression. The simplified expression is \( 16a^{20} \).

Key Concepts

Power of a Power

When dealing with exponents, a key concept to understand is the "power of a power" property. This property is particularly useful when you have an exponent raised to another exponent. The rule states that

• if you have an expression of the form \((a^m)^n\), it simplifies to \(a^{m\cdot n}\).

This means you multiply the exponents together to get a single exponent.

In the original problem, the base of the variable \(a^5\) is raised to the power of \(4\). Using the power of a power property, you multiply the exponents: \(5 \cdot 4 = 20\). Therefore, \((a^5)^4\) simplifies to \(a^{20}\).

This property makes it much easier to handle large exponents without manually expanding and multiplying the expression each time you encounter it. Just remember, only use this rule when you're dealing with an exponent raised to another exponent!

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing expressions to their simplest form while retaining their original value. The goal is to make expressions as simple and as readable as possible. When simplifying, there are a few key steps to follow:

• Identify and combine like terms.
• Apply algebraic identities and properties, like distributing exponents and applying the power of a power property.
• Simplify constants or coefficients wherever possible.

In the exercise, the expression \((2a^3a^2)^4\) was first simplified inside the parentheses. The terms with the same base \\(a^3a^2\) were combined by adding their exponents: \(a^{3+2} = a^5\). Then the outer power had to be applied to both separate components, \(2\) and \(a^5\). So, the expression inside the parentheses simplified to \(2a^5\) before continuing with the next steps. These actions ensure that the expressions are simple, consistent, and ready for further operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Understanding them is a critical component of algebra. In practice, algebraic expressions can look like:

• Simple expressions, like \(2x + 3\).
• More complex expressions, like \((2a^3a^2)^4\).

The complexity of algebraic expressions varies based on the operations involved, such as addition, subtraction, multiplication, division, and exponentiation.

In our example, the expression \((2a^3a^2)^4\) was a compound expression involving multiplication and exponentiation. Inside the parentheses were both a coefficient \(2\) and a base \(a\) raised to an exponent, where you're expected to perform operations such as exponent addition, multiplication, and distribution over each term.

To master algebraic expressions, it's important to practice recognizing patterns and applying properties and rules consistently. Each step is a building block, helping you solve for variables or simplify expressions in a manageable and logical manner.