Problem 29 - Elementary Algebra

Question

\((3 a)^{2}\)

Step-by-Step Solution

Answer

The result is \( 9a^2 \).

Step 1 - Identify the Expression

The given expression is \( (3a)^2 \). This means that we need to square the entire term inside the parentheses.

Step 2 - Apply the Power Rule

Using the Power Rule of exponents, \( (xy)^n = x^n \times y^n \), apply the exponent to both the constant and the variable inside the parentheses. Here, \( x = 3 \) and \( y = a \), and \( n = 2 \).

Step 3 - Square the Constant

Square the constant 3: \( 3^2 = 9 \).

Step 4 - Square the Variable

Square the variable \( a \: \: a^2 \).

Step 5 - Combine the Results

The squared constant and squared variable are combined back together to give the final expression: \( 9a^2 \).

Key Concepts

Power Rule

In algebra, the Power Rule is a fundamental principle for simplifying expressions involving exponents. The rule states that \( (xy)^n = x^n \times y^n \). This means when you raise a product inside parentheses to a power, you apply the exponent to each factor in the product separately.

• For example, in the expression \( (3a)^2 \), both the constant 3 and the variable \(a\) are raised to the power of 2.
• When applying the Power Rule to \( (3a)^2 \), it becomes \( 3^2 \times a^2 \).

Utilizing this principle correctly allows you to break down complex expressions into simpler ones, making it easier to solve algebraic problems. Make sure to apply the exponent to both numbers and variables separately.

Squaring

Squaring is the process of multiplying a number by itself. This operation is denoted by raising the number to the power of 2.

• For constants, squaring involves simple arithmetic. For instance, \( 3^2 = 3 \times 3 = 9 \).
• For variables, squaring follows the same principle. For example, the square of \(a\) is \(a^2\).

To square the entire term \((3a)\), apply the squaring operation to each part:

• The constant 3 is squared to get 9.
• The variable \(a\) is squared to get \(a^2\).

Combining these results gives you the term \(9a^2\). Understanding how to square both constants and variables is essential for simplifying expressions and solving equations in algebra.

Combining Terms

After applying the Power Rule and squaring each part of an expression, the next step is to combine the results.

• With the expression \((3a)^2\), you first square the constant (3) to get 9.
• Then, you square the variable \(a\) to get \(a^2\).

The final step involves combining these two results:

• You multiply the squared constant (9) by the squared variable \(a^2\).
• This gives you the expression \(9a^2\).

Combining terms correctly is crucial because it ensures the final simplified form of the expression is accurate. Always remember to check that every component has been squared and combined properly to avoid mistakes.