Problem 66 - Algebra 1: Concepts and Skills

Question

$$ \left(\frac{1}{3} m\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{1}{9} m^{2}\)

Step 1: Apply the Exponent to the Fraction

Start with applying the square to the fraction \( \frac{1}{3} \). This results in: \( \left(\frac{1}{3}\right)^{2} \) which equals \( \frac{1}{9} \).

Step 2: Apply the Exponent to the Variable m

Apply the square to the variable m, which results in: \( m^{2} \).

Step 3: Combine the Results

Combine the results of the previous steps. Therefore, the result of \( \left(\frac{1}{3} m\right)^{2} \) is \( \frac{1}{9} m^{2} \).

Key Concepts

Square of a Fraction

When you square a fraction like \( \left(\frac{1}{3}\right)^{2} \), you're multiplying the fraction by itself. This means you take \( \frac{1}{3} \times \frac{1}{3} \). Doing this involves squaring both the numerator and the denominator separately:

• The numerator is 1. Squaring 1 results in 1 because \( 1 \times 1 = 1 \).
• The denominator is 3. Squaring 3 gives you \( 3 \times 3 \), which is 9.

So, \( \left(\frac{1}{3}\right)^{2} = \frac{1}{9} \).
Squaring a fraction makes the fraction smaller, as the number moves closer to 0. This is because both the numerator and denominator grow, but the denominator grows more.

Variable Exponentiation

Variable exponentiation involves applying the exponent to any variables within the expression. In the expression \( \left(\frac{1}{3} m\right)^{2} \), the variable \( m \) is raised to the power of 2.
Here's how it works:

• The base is \( m \). When you square \( m \), you're simply multiplying \( m \) by itself.
• This results in \( m^2 \). Similarly, if \( m \) was raised to a power of any other number, you'd multiply \( m \) by itself that many times.

Applying exponents to variables follows the same rules as numbers—only the variable is repeated the number of times dictated by the exponent.
This can help simplify algebraic expressions or solve equations.

Algebraic Expressions

An algebraic expression can include numbers, variables, and operations (like addition, subtraction, multiplication, and division). The expression \( \left(\frac{1}{3} m\right)^{2} \) combines numbers and a variable in a single expression.

• This expression involves a constant (\(\frac{1}{3}\)), a variable \( m \), and an operation (squaring).
• Squaring distributes over both the constant and the variable, thereby leading to \( \frac{1}{9} m^{2} \).

Understanding how to manipulate algebraic expressions is key to solving many mathematical problems. This includes applying properties like distributive, associative, and commutative laws, to simplify or solve expressions and equations.
With practice, handling algebraic expressions becomes easier and more intuitive.