Problem 40 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ p^{2} \text { when } p=2.5 $$

Step-by-Step Solution

Answer

The result of evaluating the expression \(p^{2}\) when \(p=2.5\) is 6.25.

Step 1: Understand the expression

The expression given is \(p^{2}\), this indicates that p needs to be squared, that is, multiplied by itself.

Step 2: Substitute the value

The variable, p, should be replaced with 2.5. The resulting expression becomes \(2.5^{2}\).

Step 3: Evaluate the expression

Now perform the calculation by multiplying 2.5 by itself as per the index notation. \(2.5 \times 2.5 = 6.25\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics, especially in evaluating algebraic expressions. Whenever you see a number with a small number as a superscript, like in the expression \(p^2\), you are dealing with exponents. The small number, known as the exponent, tells you how many times to multiply the base by itself. Hence, \(p^2\) means you multiply \(p\) by itself once: \(p \times p\). For our original exercise, the base is \(p = 2.5\) and the exponent is 2.
This means the problem asks us to compute \(2.5^2\), which simplifies into a simple multiplication task.

The Method of Substitution

Substitution is the process of replacing a variable in an algebraic expression with a specific given number. It is a pivotal skill in algebra and helps transform an expression into a numerical form that can be easily evaluated. In the original exercise, the variable \(p\) is given the value 2.5.

• The expression \(p^2\) requires substituting \(p\) with 2.5.
• After substitution, the expression becomes \(2.5^2\).

Using substitution effectively allows you to simplify problems and perform calculations accurately by working directly with numbers rather than abstract variables.

Practicing Multiplication with Powers

Once you've substituted the variable, the next step often involves multiplication, especially when dealing with expressions raised to a power. In the context of our problem, we need to multiply the base \(2.5\) by itself because the exponent is 2. Here, it's crucial to perform accurate multiplication:

• Begin by aligning decimals properly if calculating by hand or use a calculator for quicker computation.
• Multiply \(2.5\) by \(2.5\).
• The result is \(6.25\).

Mastering multiplication is essential not only for this exercise but also for many areas in mathematics, as it is a foundational arithmetic operation used in various applications.

Building Confidence in Mathematics Education

Mathematics education plays a crucial role in developing problem-solving skills and logical thinking. By practicing concepts like exponents, substitution, and multiplication regularly, students build a strong mathematical foundation. This particular exercise demonstrates how different mathematical concepts interconnect:

• Understanding the role of exponents expands your knowledge of powers and roots.
• Substitution is a common technique in algebra, simplifying complex expressions.
• Multiplication ties together many math problems, reinforcing arithmetic skills.

As you continue to explore these topics, your confidence grows. Tackling each part patiently and systematically ensures a deeper understanding and helps prevent errors. Regular practice strengthens skills and fosters an appreciation for the logical beauty of mathematics.