Problem 54 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ 10-5^{2} $$

Step-by-Step Solution

Answer

-15

Step 1: Identify the operation to be performed first

According to the order of operations (PEMDAS/BODMAS), evaluate exponents before addition and subtraction. Here, the exponentiation is the first operation to perform in the expression.

Step 2: Calculate the exponentiation

Evaluate the exponentiation in the expression:\[5^2 = 5 \times 5 = 25\]So, the expression now becomes:\[10 - 25\]

Step 3: Perform the subtraction

Subtract 25 from 10:\[10 - 25 = -15\]

Key Concepts

Exponentiation

Exponentiation is a fundamental operation in mathematics. It involves raising a number, called the base, to the power of an exponent. When we see \(\text{base}^{\text{exponent}}\), it means the base is multiplied by itself as many times as the value of the exponent. For example, in \(5^2\), 5 is the base, and 2 is the exponent. \(5^2\) means 5 multiplied by itself, which gives us 25.\\The power of exponentiation becomes especially important when dealing with more complex expressions. It allows simplification and makes large calculations manageable.\\Whenever you encounter an exponent in an expression, remember that it needs to be addressed before moving on to addition and subtraction. This is where PEMDAS comes into play, ensuring that calculations follow a logical order.

PEMDAS

PEMDAS is an acronym that helps us remember the order of operations in mathematical expressions. It stands for: \\\Parentheses\Exponents\Multiplication and Division (from left to right)\Addition and Subtraction (from left to right)\When working through a problem, it's essential to follow these steps to get the correct result.\\In our example, \(10 - 5^2\), we first identify the exponentiation because 'E' comes before 'A' and 'S' in PEMDAS.\\Here's a quick breakdown of our example: \\\\Start with \(10 - 5^2\).\Evaluate the exponent: \(5^2 = 25\).\Perform the subtraction: \(10 - 25\).\The final result is -15.\ Understanding PEMDAS ensures you can effectively approach any math problem with multiple operations, giving you a reliable method to get the right answer.

Subtraction

Subtraction is one of the basic arithmetic operations. It involves taking one number away from another. In our example, we have \(10 - 25\). Here's how we approach it:\\\\Write down the numbers: 10 and 25.\Since 25 is larger than 10, we will get a negative result.\Subtract 25 from 10: 10 - 25.\The answer is -15, a negative number.\\Negative numbers result when we subtract a larger number from a smaller one. Remembering how to handle basic operations like subtraction is crucial for solving more complex problems.\\It's important to keep track of the signs (positive or negative) when performing arithmetic operations to avoid mistakes.\\Incorporating subtraction into your skill set helps you deconstruct and solve various mathematical expressions efficiently.