Question

Evaluate the expression. $$2 \cdot 3^{2} \div 7$$

Short answer

The result of the expression \(2 \cdot 3^{2} \div 7\) is approximately 2.57.

Step-by-step solution

Solve the exponent

Start by solving the exponent. So, calculate \(3^{2}\) which equals 9. The expression now reads \(2 \cdot 9 \div 7\).

Proceed with multiplication

Next, multiply 2 and 9 according to the BIDMAS/BODMAS rule. The resulting value is 18. The expression now reads \(18 \div 7\).

Finish with division

Finally, divide 18 by 7. This results in a decimal value of approximately 2.57 when rounded to two decimal places.

Key concepts

Order of Operations

Understanding the order of operations is essential when you aim to evaluate expressions accurately. The order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides you on which calculations to perform first to get the correct answer.

For instance, in the expression \(2 \cdot 3^{2} \div 7\), you start with the exponent \(3^{2}\) because exponents come before multiplication and division in the order of operations. If you didn't follow this order and multiplied or divided first, you'd end up with an incorrect answer. Always remember, if an expression contains parentheses, these should be dealt with even before exponents.

Exponents and Powers

Exponents, also known as powers, are a shorthand way to express repeated multiplication of the same number. For example, \(3^{2}\) means \(3 \times 3\), which equals 9. It's critical to solve the exponent part of an expression early on because it can significantly alter the result depending on its value.

When working with expressions like \(2 \cdot 3^{2} \div 7\), evaluating the exponent first simplifies the expression, making it easier to move through the remaining arithmetic operations. If you have an expression with more than one exponent, you should evaluate each exponent separately before proceeding with any other operations.

Arithmetic Operations

Once exponents are taken care of, you proceed with the other arithmetic operations: multiplication and division (which are of equal priority and should be carried out from left to right), followed by addition and subtraction (also equal priority, performed from left to right). In the example \(2 \cdot 3^{2} \div 7\), after calculating \(3^{2}\), you would multiply this result by 2, and then divide that product by 7. If you had additional steps that included addition or subtraction, those would come last unless grouped by parentheses.

Remember, multiplication and division do not have to be carried out strictly in one order over the other. Instead, it's based on whichever comes first from left to right in the expression. Also, if you’re dividing and multiplying several numbers, handle each operation one step at a time to avoid confusion and ensure accuracy.