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Cheat Sheet8th GradeLevel 2 · Standard

Free 8th Grade Laws of Exponents Cheat Sheet

A one-page reference for the 8th grade laws of exponents: multiplying and dividing powers with the same base, raising a power to a power, the zero index and powers of a product. Each law has a numeric and an algebraic example, and the worked examples show how to handle coefficients alongside the exponents.

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Grade
8th Grade
Subject
Math
Topic
Exponents & Scientific Notation
Difficulty
Level 2 · Standard
Estimated time
15 minutes
Curriculum
Texas grade levels (TEKS)
Answers
Not applicable
Format
PDF (A4) + print

Students will practice

  • understanding base and index notation
  • understanding and applying the multiplication, division and power-of-a-power laws
  • understanding why any non-zero number to the power of zero equals 1
  • simplifying algebraic terms with coefficients and exponents

Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Laws of Exponents Cheat Sheet

  1. 18th Grade Laws of Exponents Worksheet — Level 1
  2. 28th Grade Laws of Exponents Worksheet — Level 2
  3. 38th Grade Laws of Exponents Worksheet — Level 3
  4. 48th Grade Scientific Notation Worksheet — Level 1

How to use this cheat sheet

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practice with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this cheat sheet for?

8th Grade students (typically ages 13–14) working on exponents & scientific notation. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 15 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try 8th Grade Laws of Exponents Worksheet — Level 1.

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Success TutoringWorksheets8th Grade · Mathematics · Indices & Scientific Notation

Laws of Exponents Cheat Sheet

What you need to know

In 25, the base is 2 and the index (also called the power or exponent) is 5. It means 2 × 2 × 2 × 2 × 2 = 32. The laws of exponents are shortcuts that let you simplify expressions without writing out all the multiplications. Every law below only works when the terms have the same base.

LawRuleNumber exampleAlgebra example
Multiplyingam × an = am + n34 × 32 = 36x5 × x3 = x8
Dividingam ÷ an = am − n57 ÷ 53 = 54y9 ÷ y2 = y7
Power of a power(am)n = am × n(23)4 = 212(k2)5 = k10
Zero indexa0 = 1 (a ≠ 0)90 = 1(7x)0 = 1
Power of a product(ab)m = ambm(2 × 5)3 = 23 × 53 = 1000(2x)3 = 8x3

Simplify 23 × 24

  1. Same base (2), so add the exponents: 3 + 4 = 7.
  2. 23 × 24 = 2[sup 7] = 128.
  3. Check by expanding: 8 × 16 = 128.

Simplify 6x5 ÷ 2x2

  1. Deal with the numbers (coefficients) first: 6 ÷ 2 = 3.
  2. Same base x, so subtract the exponents: 5 − 2 = 3.
  3. 6x5 ÷ 2x2 = 3x[sup 3].

Simplify (3a2)3

  1. The power outside applies to everything inside: 33 and (a2)3.
  2. 33 = 27. For the power of a power, multiply the exponents: 2 × 3 = 6.
  3. (3a2)3 = 27a[sup 6].

Why is 50 equal to 1?

  1. Use the division law on 54 ÷ 54: subtract the exponents to get 54 − 4 = 50.
  2. But any number divided by itself is 1: 625 ÷ 625 = 1.
  3. Both answers must be the same, so 5[sup 0] = 1. The same argument works for any non-zero base.

Terms with more than one letter. Handle the coefficients first, then each base separately. 4m2n × 3mn3: coefficients 4 × 3 = 12; m2 × m = m3 (remember m means m1); n × n3 = n4. Result: 12m3n4.

Common mistake: multiplying the exponents when multiplying terms. 23 × 24 = 27, not 212. You only multiply exponents for a power of a power, such as (23)4.
Common mistake: applying the laws to different bases. 23 × 32 cannot be simplified with laws of exponents; just evaluate it: 8 × 9 = 72. Also remember that 32 means 3 × 3 = 9, not 3 × 2 = 6.

Check yourself

  1. Simplify 72 × 75.
  2. Simplify x9 ÷ x3.
  3. Simplify (52)3.
  4. Evaluate 120.
  5. Simplify 4a3 × 5a2.
  6. Simplify (2y4)3.

Answers: 1. 77 2. x6 3. 56 4. 1 5. 20a5 6. 8y12

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About this resource

Created by
Success Tutoring
Last reviewed
1 October 2026
How it was made
Written by our team

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