Problem 46 - Elementary Algebra Within Reach

Question

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(a^{2}+2 a b\) (a) \(a=-2, b=3\) (b) \(a=4, b=-2\)

Step-by-Step Solution

Answer

The result of evaluating the algebraic expression for the pairs of values are: \n(a) -8 \n(b) 0

Step 1: Evaluate the Expression for the first pair of values

Substitute \(a = -2\) and \(b = 3\) into the expression \(a^2 + 2ab\). This results in \((-2)^2 + 2*(-2)*3\). According to BIDMAS/BODMAS, carry out the power and multiplication operations before the addition, resulting in \(4 - 12\). The final step would be to perform the subtraction, resulting in a value of \(-8\) for the first pair of variables.

Step 2: Evaluate the Expression for the second pair of values

Substitute \(a = 4\) and \(b = -2\) into the expression \(a^2 + 2ab\). This results in \(4^2 + 2*4*(-2)\). As before, according to BIDMAS/BODMAS, carry out the squares operation before the multiplication and finally the addition operation. Thus, the correct sequence would be \(16 - 16\), which give a result of \(0\) for the second pair of variables.

Key Concepts

BIDMAS/BODMAS rules

When evaluating algebraic expressions, it's important to follow the order of operations to get the correct result. This is where the BIDMAS/BODMAS rules come into play. BIDMAS stands for:

• Brackets
• Indices (or Orders, which refers to exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

BIDMAS helps in remembering the sequence in which operations need to be carried out. If there are brackets in an expression, you should solve what's inside them first. Next, handle any indices by calculating powers or roots. Then, perform multiplication and division as they appear from left to right. Finally, complete any addition or subtraction as they appear from left to right too.

This rule ensures consistency and accuracy, which is crucial when dealing with complex mathematical problems. By following these steps, mistakes are minimized, allowing for an orderly approach to problem-solving.

substitution in algebra

Substitution is a fundamental skill in algebra where you replace variables with actual numerical values to evaluate expressions. When you come across an algebraic expression like \(a^2 + 2ab\), and specific values are provided, substitution is what you perform first.

Let's take the example where \(a = -2\) and \(b = 3\):

• Replace \(a\) with \(-2\), so you now have \((-2)^2\)
• Replace \(b\) with \(3\), giving \(2 \times (-2) \times 3\)

Substitution allows you to convert abstract algebraic expressions into manageable numerical calculations. It's a way of putting the math into action by transforming variables into constants and is essential in solving equations and simplifying expressions.

exponents in algebra

Exponents, also known as indices, are a shorthand way of expressing repeated multiplication. In algebra, understanding exponents is crucial since they frequently appear in expressions and equations.

For example, in the expression \(a^2 + 2ab\), \(a^2\) tells you to multiply \(a\) by itself. If \(a = 4\), then \(a^2 = 4 \times 4\), which equals 16. Similarly, if \(a = -2\), then \((-2)^2 = (-2) \times (-2)\), which equals 4.

It’s important to remember that calculating exponents comes before multiplication and addition when using the BIDMAS/BODMAS rules. This ensures that the expression is solved in the correct order to avoid errors. Understanding how to work with exponents allows you to simplify and solve equations more effectively.