Problem 49 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 3^{2}+7 $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$3^2 + 7$$ using the order of operations. Answer: 16

Step 1: Evaluate the exponent

First, we need to evaluate the exponent, which means we will calculate $$3^2$$. This represents 3 multiplied by itself two times: $$ 3^2 = 3 \times 3 = 9 $$

Step 2: Add the result to 7

Now that we have found the value of $$3^2$$, we can replace it in the original expression and add 7: $$ 9 + 7 = 16 $$ Thus, after simplifying the expression using the order of operations, the solution is: $$ 3^2 + 7 = 16 $$

Key Concepts

Exponents

Exponents are a valuable tool in simplifying and understanding mathematical expressions. They are used to denote repeated multiplication. In the expression \(3^2\), the exponent is "2" and tells us to multiply the base, which is 3, by itself. So, \(3^2\) becomes \(3 \times 3\). This type of notation helps to concisely represent large numbers or operations that require repeated multiplication.

When solving expressions with exponents, always start by evaluating any exponents present. This ensures you simplify correctly, respecting the order of operations. Exponents have a high priority in calculations and should be addressed early in the process. This step helps reduce the complexity of the equation, making further operations easier to handle. Remember, practice with exponents will solidify your understanding and improve your ability to simplify expressions efficiently.

Arithmetic Operations

Arithmetic operations include basic mathematical processes such as addition, subtraction, multiplication, and division. They are the building blocks of math and are essential for evaluating expressions. Correctly applying these operations according to specific rules, like the order of operations, is vital to solving equations accurately.

In the original problem, after dealing with the exponent, you were left with a simpler arithmetic operation: addition. Adding the previously calculated value 9 to 7 gives you 16, completing the simplification process. Always remember to perform arithmetic operations sequentially and in accordance with priority rules after reducing any more complex components, such as exponents in the equation.

Mathematical Expressions

Understanding mathematical expressions is key to solving equations successfully. A mathematical expression combines numbers, operands, and sometimes variables. In the context of using the order of operations, simplifying expressions involves breaking down a problem into manageable steps, applying the rules logically.

The expression \(3^2 + 7\) involves an exponent followed by an addition operation. Following the order of operations, sometimes called "PEMDAS" (which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), you start by evaluating the exponent. Then, proceed with simpler arithmetic processes once each component has been simplified individually.

Mathematical expressions are like puzzles. By following the correct steps and prioritizing operations, you can simplify complex problems into clear, understandable solutions.