Problem 68 - Prealgebra

Question

Find the value of each expression. $$27^{2}$$

Step-by-Step Solution

Answer

The value of \(27^2\) is 729.

Step 1: Understand the Problem

The expression given is \(27^2\). This means we need to compute the square of 27, which involves multiplying the number by itself.

Step 2: Multiply the Number by Itself

To find \(27^2\), multiply 27 by 27. \(27 \times 27 = 729\).

Step 3: Verify the Calculation

To ensure the calculation is correct, break it down further using distributive property. \(27 \times 27 = (20+7) \times (20+7) = 20^2 + 2\cdot20\cdot7 + 7^2 = 400 + 280 + 49 = 729\). Thus, \(27^2 = 729\) is accurate.

Key Concepts

Understanding Square Numbers

A square number, or perfect square, is a result of multiplying a number by itself. In our exercise, the goal is to find the square of 27, represented as \(27^2\). This simply means calculating \(27 \times 27 = 729\). Square numbers are important as they frequently appear in various areas of math, including geometry where they represent the area of a square with sides of the given number. Knowing how to calculate square numbers is a foundational math skill, which makes it easier to deal with algebraic equations, especially those involving squared terms.

Mastering the Distributive Property

The distributive property is a helpful algebraic rule that allows one to break down multiplication problems into simpler parts. This property states that \(a(b+c) = ab + ac\). In the expression \(27^2 = (20+7)^2\), we apply the distributive property as follows:

• First, calculate \(20^2\), which equals 400.
• Next, calculate \(2 \cdot 20 \cdot 7\), giving us 280.
• Finally, calculate \(7^2\), which equals 49.

Adding these values together \(400 + 280 + 49\), we once again get 729. Utilizing the distributive property simplifies complex multiplications and ensures accuracy in mathematical calculations. This method is particularly useful for squaring larger numbers.

Boosting Your Multiplication Skills

Strong multiplication skills are essential for tackling a variety of math problems, including those involving exponents and squares. In our problem of finding \(27^2\), multiplying 27 by itself directly gives us 729. Developing these skills relies on understanding number relationships and practicing regularly. Consider these tips:

• Use the distributive property to make multiplication easier, as seen in breaking down \(27 \times 27\) into \((20+7)^2\).
• Practice multiplication tables to increase speed and confidence.
• Break down larger numbers into sums or differences that are easier to multiply.

By improving your multiplication skills, you set a solid foundation for tackling not only square numbers but also more advanced topics in mathematics with ease.