Laws of exponents – Grade 11 math lesson

Lesson 4 of 23 in the Grade 11 math course

US version. US grades, US spelling and dollars ($). · UK Maths Department

A complete 45-minute lesson for grade 11 (ages 16–17) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.

Time: 45 minutes + 20 min homework
Grade: Grade 11 (ages 16–17)
You need: Paper, a pencil and this page
Graded: Instantly, with explanations

For parents: how to use this lesson

You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is graded straight away with a worked explanation, so you never have to grade anything.

Read what your child will learn and the big ideas together.
Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
Finish with the exit ticket and the talk question: “Why does 5⁰ equal 1? Use the division law to explain.”
Later that day or the next day: the homework (20 minutes), then a practice test.

Watch out for

Multiplying the indices when multiplying powers (2³ × 2⁴ = 2¹²). Write it out: three 2s times four 2s is seven 2s.
Thinking 3⁻² is negative. A negative index makes a fraction: 1/9.

Tip for home: Investigate how quickly doubling grows: 2¹⁰, 2²⁰.

What your child will learn

Learning goal: use the laws of exponents, including zero, negative and fractional exponents.

Success looks like

I can multiply and divide powers.
I can use zero and negative indices.
I can use fractional indices.

Key words

exponent: the power a number is raised to
base: the number being raised to a power
reciprocal: 1 divided by the number
root: the opposite of a power

What to have ready

Paper and a pencil
This page on a computer, tablet or phone
For the hands-on part: Write out powers in full (2 × 2 × 2) to discover the laws for yourself.

The big ideas (for the adult)

When multiplying powers of the same number, add the indices: 2³ × 2⁴ = 2⁷. When dividing, subtract them.

A power of a power multiplies the indices: (5²)³ = 5⁶. Anything to the power 0 is 1.

A negative index means a reciprocal: 3⁻² = 1/9. A fractional index means a root: 8^(1/3) = 2.

The lesson, step by step

45 minutes in 5 parts
Part	Time	How it is checked
Starter: quick recall	5 min	Graded by Classfern
Teaching input	10 min	Not assessed, self-checked
Worked examples and guided practice	9 min	Not assessed, graded by Classfern
Independent practice	15 min	Graded by Classfern
Closure and exit ticket	6 min	Graded by Classfern, self-checked

Starter, "your turn", practice and exit ticket are graded by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.

The lesson

It runs here on the page. No sign-up and no names – progress is kept only in this browser.

Homework and practice at home

20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all graded instantly.

More practice

All Grade 11 math lessons and tests · Math Department

Curriculum

Common Core State Standards HSN.RN.A.2: “Rewrite expressions involving radicals and rational exponents using the properties of exponents.”

Common Core State Standards HSN.RN.A.1: “Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))³ = 5^((1/3)3) to hold, so (5^(1/3))³ must equal 5.”

Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.