Problem 28 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (2 x)^{2} $$

Step-by-Step Solution

Answer

The expression \((2x)^2\) can be written as: \(4x^2\).

Step 1: Apply the power of a power rule

Power of a power rule states that when you have a power of a power, like \((a^n)^m\), you multiply the exponents together. Here, we have \((2x)^2\), which means 2 and x are both to the second power. So, this expression can be written as \(2^2 \cdot x^2\).

Step 2: Simplify the numerical exponent part

Now, \(2^2\) simply equals 4.

Step 3: Write final result

Putting it all together, the expression \((2x)^2\) gets simplified to \(4x^2\).

Key Concepts

Power of a Power Rule

The "power of a power" rule is a fundamental concept in exponents. It is useful when dealing with expressions like \((a^n)^m\). The rule states that you multiply the exponents: \((a^n)^m = a^{n \cdot m}\). This rule simplifies the processes involved in dealing with exponents by reducing potential multiple steps into one simple calculation.

If you have an expression such as \((2x)^2\), you apply the power of a power rule to both the base 2 and the variable \(x\).
Here's how it looks:

• The number 2 is raised to the power of 2: \(2^2\).
• The variable \(x\) is also raised to the power of 2: \(x^2\).

Effectively, each component within the parentheses is affected by the external power, simplifying the expression to \(2^2 \cdot x^2\).

This step is crucial for further simplification.

Simplifying Expressions

Once the exponents are correctly applied using the power of a power rule, the next step is to simplify the expression.

Simplification involves reducing the expression to its most basic form. In our example, \((2x)^2\), we've already rewritten it as \(2^2 \cdot x^2\). Now, we focus on simplifying the numerical exponent, \(2^2\), which is straightforward:

• \(2^2\) calculates to 4, since you're multiplying 2 by itself: \(2 \times 2 = 4\).

This leads us to an expression \(4 \cdot x^2\).

Each step in simplifying expressions is important as it helps us condense complex mathematical statements into a more manageable and interpretable form. This is not only useful but necessary for solving algebraic problems efficiently.

Algebraic Expressions

Understanding algebraic expressions is vital for mastering algebra. An algebraic expression involves numbers, variables, and operations (like addition and multiplication). In our workings, \((2x)^2\) is an algebraic expression.

Algebraic expressions can often seem intimidating because they combine numbers with letters, but they provide a powerful means to represent and solve real-world problems. Here’s a breakdown:

• The expression \((2x)^2\) involves the number 2 and the variable \(x\), both under an exponent.
• It shows how numbers and variables can be manipulated by laws of exponents, like the power of a power rule, to solve expressions.

Writing algebraic expressions in their simplest forms is essential. It reduces errors and aids in understanding relationships between different elements of the expressions.

Whenever you simplify expressions like \(4x^2\), you are leveraging the power of algebra to make complex relationships simpler and clearer. This deepens your understanding and enhances problem-solving skills.