Skills Beta

Formulas, Order of Operations and Square Roots

Evaluate a formula by substituting values, then following the order of operations: grouped terms first (brackets, fraction bars, roots), then powers, then multiplication and division, then addition and subtraction.

Practice 3: Analyze Data

Part 1 · Hook

Why this matters

In the next topic you will meet formulas like ∑(xi−5)2n−1\sqrt{\frac{\sum (x_i - 5)^2}{n - 1}}. They look fierce, but each is a short recipe: subtract, square, add, divide, take a root. Doing the steps in the right order is the whole skill, and a calculator only helps if you type the brackets it needs.

Part 2 · Prerequisite check

Quick check before you start

1. What is 1824\frac{18}{24} as a decimal?

  1. 0.75
  2. 1.33
  3. 0.18
  4. 0.6
Show the answer

Divide the top by the bottom: 18 ÷ 24 = 0.75.

  • Correct: 0.75:
  • 1.33:
  • 0.18:
  • 0.6:

2. A distribution has a long tail toward larger values. What is its shape?

  1. Skewed right
  2. Skewed left
  3. Symmetric
  4. Uniform
Show the answer

Skew is named for the tail.

  • Correct: Skewed right:
  • Skewed left:
  • Symmetric:
  • Uniform:

Part 3 · See it

See it first

Three values, 2, 4 and 9, as dots on a number line from 0 to 10. A plum line marks their average, (2 + 4 + 9) / 3 = 5. Arrows from the average to each value show the distances 2 − 5 = −3, 4 − 5 = −1 and 9 − 5 = 4.
The values 2, 4 and 9 with their average, 5. The distances from the average are −3, −1 and 4. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

  1. A formula is a recipe written in symbolssubstitute the numbers for the letters first
  2. Brackets and fraction bars group termswork out what is inside them before anything else
  3. Then powers and roots, then multiplication and division, then addition and subtractionevery reader of the formula gets the same answer
  4. A square root or fraction bar covers a whole expressiontype it on a calculator with brackets around everything it covers

Part 5 · Key ideas

Key ideas

  • Order of operations: brackets (and anything grouped by a fraction bar or root) first, then powers and roots, then multiplication and division, then addition and subtraction.
  • A square root a\sqrt{a} is the nonnegative number whose square is aa: 0.25=0.5\sqrt{0.25} = 0.5. Squares are never negative: (−3)2=9(-3)^2 = 9.
  • Summation: ∑xi\sum x_i says add every value; ∑(xi−5)2\sum (x_i - 5)^2 says subtract 5 from each value, square each result, then add.
  • Show the work the way the exam asks: the formula, the numbers substituted, then the answer.

Part 6 · Misconception

A common mistake

The wrong idea: 2+4+9/3=52 + 4 + 9/3 = 5, because that is the average of 2, 4 and 9.

What actually happens: Without brackets only the 9 is divided: 2+4+3=92 + 4 + 3 = 9. The average is (2+4+9)/3=5(2 + 4 + 9)/3 = 5; the fraction bar groups the whole top.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. Evaluate 2+4+93\frac{2 + 4 + 9}{3}.

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

The fraction bar acts like brackets: add the top first, 2+4+9=152 + 4 + 9 = 15, then divide, 153=5\frac{15}{3} = 5.

  • Answer: 5

2. Evaluate (2−5)2+(4−5)2+(9−5)23−1\sqrt{\frac{(2 - 5)^2 + (4 - 5)^2 + (9 - 5)^2}{3 - 1}}. Give two decimal places.

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

Inside first: (−3)2(-3)^2+(−1)2+ (-1)^2+42+ 4^2=9= 9+1+ 1+16+ 16=26= 26; the bottom is 3−1=23 - 1 = 2; 262=13\frac{26}{2} = 13; then 13=3.61\sqrt{13} = 3.61.

  • Answer: 3.61

3. For the values x1=3x_1 = 3, x2=5x_2 = 5 and x3=10x_3 = 10, what is ∑xi\sum x_i?

  1. 150
  2. 18
  3. 6
  4. 3
Show the answer

∑xi\sum x_i tells you to add every xix_i: 3+5+10=183 + 5 + 10 = 18.

  • 150: The summation sign adds; it does not multiply.
  • Correct: 18: Right: the summation sign adds every value, 3+5+10=183 + 5 + 10 = 18.
  • 6: Dividing by 3 gives the average; the sum itself is 18.
  • 3: That is how many terms there are, not their sum.

4. Type an expression for the average of 12, 15, 18 and 21: add the values, then divide by how many there are.

Type the expression as on the exam, with the values substituted.

Show the answer

12+15+18+214\frac{12 + 15 + 18 + 21}{4}=664= \frac{66}{4}=16.5= 16.5.

  • Answer: 12+15+18+214\frac{12 + 15 + 18 + 21}{4}

5. What is (−3)2(-3)^2?

  1. 9
  2. −9
  3. −6
  4. 6
Show the answer

A square is never negative: (−3)2=9(-3)^2 = 9. On a calculator, type the brackets, or −32-3^2 gives −9-9.

  • Correct: 9: Right: (−3)×(−3)=9(-3) \times (-3) = 9; a negative times a negative is positive.
  • −9: That is −(32)-(3^2). With the brackets, the whole −3-3 is squared.
  • −6: Squaring multiplies a number by itself, not by 2.
  • 6: A square is a number times itself: (−3)(−3)(-3)(-3).

6. Type an expression for the square root of the whole quantity 26 divided by 2.

Type the expression as on the exam, with the values substituted.

Show the answer

262=13≈3.61\sqrt{\frac{26}{2}} = \sqrt{13} \approx 3.61. The brackets of the root hold the whole fraction.

  • Answer: 262\sqrt{\frac{26}{2}}

Part 8 · Summary

Summary

Evaluate a formula by substituting values, then following the order of operations: grouped terms first (brackets, fraction bars, roots), then powers, then multiplication and division, then addition and subtraction. ∑\sum adds a list of terms, and a square root covers everything under it, so type brackets on a calculator.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Share to Google Classroom (opens in a new tab)

Question set for this topic