Problem 41 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-2[5 a+2 b(b-6)], \text { if } a=-2 \text { and } b=3$$

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1: Substitute the values

First, we substitute the given values of the variables into the expression. We have \( a = -2 \) and \( b = 3 \). So, replace \( a \) with \( -2 \) and \( b \) with \( 3 \) in the expression: \[ -2 (5 (-2) + 2 (3)(3 - 6)) \]

Step 2: Simplify the expression inside the brackets

Next, we simplify the expression inside the brackets. Start by calculating the parts of the expression separately: \( 5(-2) = -10 \) and for the second part:\( 2(3(3 - 6)) = 2(3 \times -3) = 2(-9) = -18 \). Now replace these in the expression:\[ -2(-10 - 18) \]

Step 3: Combine the terms inside the brackets

Add the results from the previous step inside the brackets: \( -10 - 18 = -28 \). Now, the expression becomes:\[ -2(-28) \]

Step 4: Calculate the final result

Multiply the numbers:\( -2 \times -28 = 56 \). This is because multiplying two negative numbers gives a positive result.

Key Concepts

Algebraic expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition, subtraction, multiplication, and division). It is a way to represent real-life scenarios using mathematical symbols. For instance, the expression \(-2[5a + 2b(b-6)]\) combines constants, numbers, and variables \(a\) and \(b\). Variables like \(a\) and \(b\) are placeholders for values we want to find or use, and algebraic expressions can vary in complexity. These expressions are key in solving equations, simplifying mathematical problems, and representing patterns or rules in a concise form. To work with these expressions effectively, it's essential to know how to substitute specific values in for the variables and simplify the expression accordingly. This requires practice with mathematical operations and an understanding of how variables function in equations.

Order of operations

When simplifying or evaluating algebraic expressions, it's crucial to follow the correct order of operations. This ensures that everyone solves the expression the same way and gets the same result. The order of operations is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When evaluating the expression \(-2[5a + 2b(b-6)]\), we first deal with the operations inside the brackets, which take precedence. Inside these brackets, multiplication is done first before the addition and subtraction. This process ensures that all operations are performed in the correct sequence, leading to the right solution.

Negative numbers multiplication

Multiplying negative numbers can be a bit tricky, but following a few simple rules makes it easier. If two numbers share the same sign (both positive or both negative), their product is positive. In contrast, if the numbers have different signs, their product is negative. This pattern occurs due to how multiplication is defined in mathematics. Take for example the expression we solved: concluding with multiplying \(-2\) and \(-28\). Both numbers are negative, so the product, \(56\), is positive. It helps to think of multiplication involving negative numbers as counting in reverse on the number line: you multiply as usual, and the signs determine the direction. Remembering these rules helps avoid confusion and ensures accurate calculations when dealing with expressions that involve negative numbers.