Order of Operations
The order of operations is the universal hierarchy of mathematical calculations: first evaluate grouping symbols (parentheses, fraction bars, radicals), then exponents and roots, then multiplication and division left to right, and finally addition and subtraction left to right. This Grade 8 math skill from Yoshiwara Core Math Chapter 4 adds the full treatment of radicals and fraction bars to the basic PEMDAS students learned earlier. Mastering order of operations ensures consistent, correct results in all mathematical calculations from arithmetic through calculus. Errors in this rule are among the most common sources of wrong answers in algebra.
Key Concepts
Property 1. First, perform all operations inside parentheses, or above or below a fraction bar, or inside a radical. 2. Next, compute all powers and roots. 3. Perform all multiplications and divisions in order from left to right. 4. Finally, perform all additions and subtractions in order from left to right.
Examples To simplify $10 + 2\sqrt{25}$, first evaluate the root: $10 + 2(5)$. Then multiply: $10 + 10$. Finally, add to get $20$.
In $\frac{10 + \sqrt{100}}{8 \sqrt{9}}$, first evaluate the roots: $\frac{10+10}{8 3}$. Then simplify the numerator and denominator: $\frac{20}{5}$. The final answer is $4$.
Common Questions
What is the order of operations?
The order of operations is: (1) grouping symbols - parentheses, fraction bars, radicals; (2) exponents and roots; (3) multiplication and division left to right; (4) addition and subtraction left to right. Always work from the inside out for nested groupings.
What does PEMDAS stand for?
PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is a memory device for the order of operations. Multiplication and division are done together left to right, as are addition and subtraction.
Why must operations inside radicals be done first?
A radical symbol (like a square root sign) is a grouping symbol. Everything under the radical must be evaluated before taking the root. For example, sqrt(144 + 25) = sqrt(169) = 13, not sqrt(144) + sqrt(25) = 12 + 5 = 17.
When do 8th graders learn the full order of operations?
Students study the complete order of operations including fraction bars and radicals in Grade 8 math as part of Chapter 4 of Yoshiwara Core Math, which covers calculation rules.
How do fraction bars work as grouping symbols?
The numerator and denominator of a fraction are each evaluated completely before the division is performed. For example, (10 + sqrt(100)) / (8 - sqrt(9)) = (10 + 10) / (8 - 3) = 20 / 5 = 4.
What is a common order of operations mistake?
A common mistake is treating a radical as applying only to the first number rather than all contents under the symbol. Another mistake is doing addition before multiplication. For example, 2 + 3 x 4 = 2 + 12 = 14, not 5 x 4 = 20.