Evaluate the expression:
$(-5)^2 \times 3 \div 5+9$
To evaluate the expression \((-5)^2 \times 3 \div 5 + 9\), we must follow the order of operations, often remembered by the acronym PEMDAS or BODMAS. This rule dictates the sequence in which we perform calculations: Parentheses, Exponents, Multiplication and Division (processed from left to right), and finally Addition and Subtraction (also processed from left to right). Our expression consists of an exponent, multiplication, division, and addition.
First, we handle the exponentiation. The term \((-5)^2\) means \((-5) \times (-5)\). Since a negative number multiplied by a negative number results in a positive number, \((-5)^2 = 25\). The expression now becomes \(25 \times 3 \div 5 + 9\).
Next, we perform the multiplication and division from left to right. We first calculate \(25 \times 3 = 75\). Then, we divide the result by 5, so \(75 \div 5 = 15\). The expression now simplifies to \(15 + 9\).
Finally, we perform the addition: \(15 + 9 = 24\). Thus, the numerical value of the given expression is 24.
The order of operations is essential in everyday life, especially in financial calculations. For example, if you are calculating the total cost of an item that has a discounted price and a tax rate, you must follow the correct sequence of operations to ensure accuracy. If you calculate the total as (Discounted Price) + (Tax Rate * Price) instead of applying the tax correctly, your final total could be wrong, which is why following the order of operations is crucial for budgeting and financial planning. In this math problem, we correctly calculated \((-5)^2 \times 3 \div 5 + 9 = \boxed{24}\).