Simplify the expression:
$2\left(4^2-3\right)-7$
We are tasked with evaluating the mathematical expression \(2(4^2 - 3) - 7\). To do this correctly, we must follow the standard order of operations, often referred to as PEMDAS or BODMAS. This rule dictates that we perform operations in the following sequence: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
First, we focus on the part of the expression inside the parentheses: \(4^2 - 3\). According to the order of operations, we calculate the exponent first. We know that \(4^2 = 4 \times 4 = 16\). Substituting this back into the parenthetical expression gives us \(16 - 3\). Performing the subtraction, we get \(16 - 3 = 13\).
Now we substitute the result from the previous step back into the original expression. The expression now becomes \(2(13) - 7\). Next, we perform the multiplication: \(2 \times 13 = 26\). Finally, we perform the subtraction: \(26 - 7 = 19\).
Order of operations is essential in everyday life, not just in math class. For example, if you are shopping and have a coupon for a discount, you must know whether to apply the discount before or after tax. Similarly, when programming a computer to perform a series of financial calculations, you must ensure the sequence is correct so that the total balance or interest is calculated accurately. Mastering these rules ensures that you handle complex tasks, like personal budgeting or scientific measurements, without errors.