Problem 77 - Algebra 1

Question

Evaluate the expression. \(5 \cdot 2 p^{2}\) when \(p=6\)

Step-by-Step Solution

Answer

The evaluated expression is \(360\).

Step 1: Identify the operations and their order

In the expression \(5 \cdot 2 p^{2}\), the operations involved are multiplication and exponentiation. According to the BODMAS rule, exponentiation should be handled before multiplication.

Step 2: Substitute value for variable

Replace \(p\) in the expression with \(6\). Now the expression would look like \(5 \cdot 2 \cdot 6^{2}\).

Step 3: Carry out exponentiation

Calculate \(6^{2}\), which results in \(36\). Substituting this back into our expression, we get \(5 \cdot 2 \cdot 36\).

Step 4: Perform multiplication

Multiply the numbers in the order they appear, from left to right. Calculating \(5 \cdot 2\) gives \(10\). Substituting this back, we get \(10 \cdot 36\). Finally, multiplying \(10\) and \(36\) gives \(360\).

Key Concepts

Order of operations

When dealing with arithmetic and algebra, it's crucial to perform operations in a specific order to obtain the correct result. This sequence is remembered by the acronym PEMDAS in the United States or BODMAS/BIDMAS in other countries. Essentially, these stand for Parentheses/Brackets, Exponents/Orders (i.e., powers and roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, the expression involves multiplication as well as exponentiation. To properly evaluate the expression, one must first take care of exponentiation — which is 'raising a number to the power of another' — before proceeding with multiplication. Skipping or jumbling this order could lead to incorrect answers and confusion. It's essential for students to follow this standard convention to avoid errors and achieve precision in their calculations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When you exponentiate, you raise the base to the power of the exponent, which means you multiply the base by itself as many times as specified by the exponent. For example, in the expression \(6^2\), 6 is the base and 2 is the exponent; this reads as '6 squared' or '6 raised to the power of 2', resulting in \(6 \cdot 6 = 36\).

Understanding exponentiation is vital because it's used extensively across different areas of mathematics, including algebra, geometry, and calculus. Students should remember that any number to the power of 0 is 1 and any number to the power of 1 is the number itself. These fundamental concepts serve as building blocks for more complex operations involving exponentiation.

Substitution in algebra

Substitution is a fundamental algebraic tool that involves replacing a variable in an expression with a specific value. This technique is particularly common when solving equations or evaluating expressions. In our example, \(p\) is the variable and the exercise instructs to substitute it with the value 6. After substitution, one can proceed with the algebraic operations as usual to find the value of the expression.

The process of substitution requires careful attention to detail to replace every instance of the variable correctly. It's also essential to ensure that the substitution is consistent throughout the entire expression. This clear, methodical approach makes complex algebraic evaluations simpler and easier to manage, enabling students to solve problems accurately and efficiently.