Unit 1 · Lesson 4 of 6

Exponents, square roots, and order of operations

About 20-25 minutes · 5 steps · one idea: the order you compute in

You'll be able to: compute exponents and square roots, and work out any arithmetic expression in the right order.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first. Getting it slightly wrong and then fixing it still helps you remember.

\(\dfrac{5}{6} - \dfrac{1}{4}\)

The LCD of 6 and 4 is 12. \(\dfrac{5}{6} = \dfrac{10}{12}\) and \(\dfrac{1}{4} = \dfrac{3}{12}\). Subtract the tops: \(\dfrac{10 - 3}{12} = \dfrac{7}{12}\).

\((-4)(-5)(-1)\)

\((-4)(-5) = 20\) (same signs, positive). Then \(20 \cdot (-1) = -20\) (different signs, negative).

Step 2 of 5The idea

An exponent is shorthand for multiplying a number by itself: \(2^3 = 2 \cdot 2 \cdot 2 = 8\). The 2 is the base and the 3 is the exponent.

A square root undoes squaring: \(\sqrt{36} = 6\) because \(6^2 = 36\). Learn the perfect squares by heart: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Order of operations.

1. Parentheses, and anything grouped by a fraction bar or a root sign.
2. Exponents and roots.
3. Multiply and divide, left to right.
4. Add and subtract, left to right.

An exponent touches only what is right next to it. \(-3^2 = -(3 \cdot 3) = -9\), but \((-3)^2 = (-3)(-3) = 9\).

A fraction bar groups. Finish the whole top and the whole bottom first, then divide.

Multiply and divide have the same rank, so go left to right: \(20 \div 5 \cdot 2 = 4 \cdot 2 = 8\). Add and subtract work the same way.

Why this shows up on the placement test: expressions like \(-x^2\) and \(b^2 - 4ac\) are everywhere, and one order-of-operations slip changes the answer.

Step 3 of 5Worked examples

Example 1: \(-4^2\), \((-4)^2\), and \(\sqrt{81}\)

  1. \(-4^2\): the exponent touches only the 4, so \(-(4 \cdot 4) = -16\).
  2. \((-4)^2\): the parentheses put the negative inside, so \((-4)(-4) = 16\). (Same signs, positive.)
  3. \(\sqrt{81} = 9\), because \(9^2 = 81\).

Example 2: \(\dfrac{20 - 3 \cdot 2^3}{\sqrt{16}}\)

  1. The fraction bar groups, so work the top and the bottom separately.
  2. Top, exponent first: \(2^3 = 8\). Then multiply: \(3 \cdot 8 = 24\). Then subtract: \(20 - 24 = -4\). (Multiply before subtract.)
  3. Bottom: \(\sqrt{16} = 4\).
  4. Divide: \(\dfrac{-4}{4} = -1\).

Step 4 of 5Practice

Work each one on paper, step by step. Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.

\(5^3\)

\(5^3 = 5 \cdot 5 \cdot 5\). First \(5 \cdot 5 = 25\), then \(25 \cdot 5 = 125\).

\(\sqrt{144} - \sqrt{49}\)

Roots first: \(\sqrt{144} = 12\) and \(\sqrt{49} = 7\). Then subtract: \(12 - 7 = 5\).

\((-3)^2 + (-2)^3\)

\((-3)^2 = (-3)(-3) = 9\). \((-2)^3 = (-2)(-2)(-2) = 4 \cdot (-2) = -8\). Then \(9 + (-8) = 1\).

Which expression is equal to \(16\)?

  1. \(-2^4\)
  2. \((-2)^4\)
  3. \(-4^2\)
  4. \(-(-4)^2\)

(a) The exponent touches only the 2: \(-(2 \cdot 2 \cdot 2 \cdot 2) = -16\).

(b) \((-2)(-2)(-2)(-2) = 16\). Four negatives multiply to a positive. This one.

(c) \(-(4 \cdot 4) = -16\).

(d) \((-4)^2 = 16\), and the minus sign in front makes it \(-16\).

\(18 - 12 \div 3 \cdot 2\)

Multiply and divide before subtracting, left to right: \(12 \div 3 = 4\), then \(4 \cdot 2 = 8\). Last, \(18 - 8 = 10\).

\(4 + 2(7 - 3)^2\)

Parentheses: \(7 - 3 = 4\). Exponent: \(4^2 = 16\). Multiply: \(2 \cdot 16 = 32\). Add: \(4 + 32 = 36\).

\(\dfrac{3^2 + 7}{10 - 2 \cdot 3}\)

Top: \(3^2 = 9\), and \(9 + 7 = 16\). Bottom: \(2 \cdot 3 = 6\), and \(10 - 6 = 4\). Divide: \(\dfrac{16}{4} = 4\).

\(\left(\dfrac{1}{2}\right)^2 + \dfrac{3}{4} \cdot \sqrt{4}\)

Exponent and root first: \(\left(\dfrac{1}{2}\right)^2 = \dfrac{1}{4}\) and \(\sqrt{4} = 2\). Multiply: \(\dfrac{3}{4} \cdot 2 = \dfrac{6}{4}\). Add: \(\dfrac{1}{4} + \dfrac{6}{4} = \dfrac{7}{4}\).

Step 5 of 5Wrap-up

You can now compute exponents and square roots and follow the order of operations, including the \(-3^2\) versus \((-3)^2\) trap and fraction bars that group.

Read or watch next: OpenStax Elementary Algebra 2e, 1.2 Use the Language of Algebra (exponents and order of operations) and 1.8 The Real Numbers (square roots).

Also good: Khan Academy: Order of operations (short videos). Quick rules: Number skills cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "Why is \(-3^2\) negative but \((-3)^2\) positive?" It knows this course and can explain things another way.