Problem 64 - Prealgebra

Question

Find the value of each expression. $$15^{2}$$

Step-by-Step Solution

Answer

The value of \(15^2\) is 225.

Step 1: Understand the Expression

The expression given is \(15^2\). This means 15 is raised to the power of 2, or in other words, 15 is multiplied by itself.

Step 2: Perform the Multiplication

To solve \(15^2\), multiply 15 by itself: \(15 \times 15\).

Step 3: Calculate the Result

Perform the multiplication: \(15 \times 15 = 225\). Therefore, \(15^2 = 225\).

Key Concepts

Squared Numbers

Squared numbers are fundamental in mathematics and easy to understand with practice. When we say a number is squared, we're referring to the number being multiplied by itself. In our example, the expression given was \(15^2\). This is sometimes read as "15 squared" and signifies \(15 \times 15\).

It's important to note that squaring is a type of exponentiation where the exponent, or power, is 2. This is why squared numbers are also known as second powers. Squaring transforms any number into a larger number unless we're dealing with fractions or decimals between 0 and 1, which become smaller when squared.

• Examples: \(3^2 = 9\), \(10^2 = 100\)
• Visual Tip: Squaring can be visualized as finding the area of a square with side length equal to the number being squared.

Multiplication

Multiplication is one of the basic operations in arithmetic and is vital when dealing with squared numbers. It's essentially repeated addition. For example, instead of adding 15 to itself repetitively (\(15 + 15 + 15...\)), multiplication allows us to compute \(15 \times 15\) directly.

When you're squaring a number, you're performing multiplication with two identical factors. In our original exercise, calculating \(15^2\) required us to multiply 15 by 15, resulting in 225.
Multiplication rules to remember:

• Commutative Property: Order doesn't matter (\(a \times b = b \times a\)).
• Associative Property: Grouping doesn't change the result ((\((a \times b) \times c = a \times (b \times c)\)).

Understanding multiplication establishes a solid foundation for more complex mathematics.

Arithmetic

Arithmetic is the branch of mathematics concerning basic numerical operations like addition, subtraction, multiplication, and division. It's the toolbox we use to understand and solve numerical problems in everyday and academic contexts.

The exercise of finding \(15^2\) is a straightforward arithmetic problem that highlights how arithmetic operations can be used to compute powers like squaring a number. This arithmetic task showcases various properties, such as pattern recognition (how numbers change when squared) and computational strategies (like breaking down complex calculations into simpler parts).
Key points for effective arithmetic:

• Practicing operations regularly improves problem-solving speed and accuracy.
• Understanding place value is critical for mitigating errors in larger computations.

Comprehending arithmetic prepares students not just for academic success but also for practical applications in real life.