Problem 1 - Algebra 1: Concepts and Skills

Question

Place the operations in the order in which you should do them. a. Multiply and divide from left to right. b. Do operations within grouping symbols. c. Add and subtract from left to right. d. Evaluate powers.

Step-by-Step Solution

Answer

The correct order of operations is: b. Do operations within grouping symbols, d. Evaluate powers, a. Multiply and divide from left to right, and finally c. Add and subtract from left to right.

Step 1: Grouping Symbols

Start with operations within grouping symbols. This is also known as brackets or parentheses in mathematical terms. Therefore, the first operation in order should be 'b. Do operations within grouping symbols.'

Step 2: Powers

The next operation should be evaluating powers. This is because, after brackets, powers or exponents are processed next. Therefore, the second operation in order should be 'd. Evaluate powers.'

Step 3: Multiply and Divide

After exponents, the next step is to perform multiplication and division from left to right. Therefore, 'a. Multiply and divide from left to right' should be the third operation in the proper order.

Step 4: Add and Subtract

The final operation should be addition and subtraction, performed from left to right. Therefore, 'c. Add and subtract from left to right' should be the last operation in the order.

Key Concepts

Grouping Symbols

Grouping symbols are a crucial component in mathematics, often taking the form of parentheses \(( )\), brackets \([ ]\), or braces \({ }\). These symbols tell us to carry out the operations they enclose first. When you see an expression with grouping symbols, treat the operations inside them as a priority.

For example, in the expression \(2 \times (3 + 4)\), you would first complete the operation inside the parentheses, which is \(3 + 4\), before performing the multiplication. This ensures that all calculations are in the intended order. Always be on the lookout for grouping symbols first, as they set the groundwork for all other calculations in the expression.

Exponents

Exponents, also known as powers, are the next in line after dealing with grouping symbols. An exponent, placed as a small number above and to the right of a base number, indicates how many times to multiply the base by itself.

For example, in the expression \(5^3\), the base is 5 and the exponent is 3, meaning you calculate \(5 \times 5 \times 5\). It is crucial to compute the exponents before moving on to other operations because they significantly impact the final result by exponentially increasing the value of the base number.

Multiplication and Division

Multiplication and division should be dealt with next, once exponents have been evaluated. The order of operations requires these to be completed from left to right, as you encounter them in the equation.

In an expression like \(4 \div 2 \times 3\), begin with the division \(4 \div 2\), yielding 2, and then multiply by 3 to get 6. It's important not to flip the order even if multiplication is before division in a different part of the equation, maintain the left-to-right approach. This helps in achieving accuracy, particularly in longer and complex mathematical expressions.

Addition and Subtraction

Finally, addition and subtraction are the last operations to perform, but just like multiplication and division, they must be completed from left to right. These are often the simplest calculations, but maintaining the correct order is key to avoiding errors.

Consider an expression like \(7 - 5 + 3\). Here, you'd first subtract \(7 - 5\) to get 2, and then add 3, reaching a final result of 5. The left-to-right rule for addition and subtraction ensures that the mathematical process follows a logical sequence, leading to the most accurate outcome in any problem.