Exponent rules and working with roots
What this lesson covers
- Exponent rules govern how operations involving exponents work and include: a^m × aⁿ = a^(m+n) (the product-of-exponents rule) a^m/aⁿ = a^(m− n) (the quotient-of-exponents rule) (a^m)ⁿ = a^(m×n) (the exponent-of-exponents rule) a^(−m) = 1/a^m, (a ≠ 0) (the negative exponent rule) a⁰ = 1 (a ≠ 0) (the zero exponent rule)
- Only like roots can be added and subtracted; multiples of a root are represented with coefficients (e.g. √3 + √3 = 2√3).
- There are rules for working with roots, including not leaving roots in a denominator: √a × √a = a √a × √b = √ab √a ÷ √a = 1 √a ÷ √b = √(a/b) = √a/√b × √b/√b = √ab/b
Five rules, or one idea five times
- An exponent counts factors. That is all it ever does.
- So multiplying counts them together, dividing cancels them.
- Every rule in this lesson comes from writing it out long.
Multiplying: add the exponents
- Two cubed is three twos. Two to the fourth is four twos.
- Put them side by side and you have seven twos.
- So the exponents added themselves. You did not make a rule.
| Power | Written out | Count |
|---|---|---|
| 2³ | 2 × 2 × 2 | three twos |
| 2⁴ | 2 × 2 × 2 × 2 | four twos |
| 2³ × 2⁴ | all of them at once | seven twos |
| So | 2³ × 2⁴ = 2⁷ | = 128 |
Dividing: subtract them
- Seven twos on top, four twos underneath.
- Four of them cancel, the way any fraction cancels.
- Three twos survive, so the exponents subtracted.
| Step | What happens | Left |
|---|---|---|
| 2⁷ ÷ 2⁴ | seven over four | |
| Cancel | four twos go | three left |
| So | 2⁷ ÷ 2⁴ = 2³ | = 8 |
| The rule | subtract the exponents |
A power of a power: multiply them
- Two cubed, all to the fourth, means four lots of two cubed.
- Each lot is three twos, and there are four lots.
- Four threes are twelve, so it is two to the twelfth.
| Step | Meaning | Result |
|---|---|---|
| (2³)⁴ | four lots of 2³ | |
| Each lot | is three twos | |
| Altogether | 4 × 3 = 12 twos | 2¹² |
| Check | 8⁴ = 4096 | 2¹² = 4096 |
The two that look invented
- Two cubed over two cubed is 1. Obviously.
- The subtract rule says it is two to the zero.
- So two to the zero is 1, and nobody decided that.
| Start from | Which gives | So |
|---|---|---|
| 2³ ÷ 2³ | is plainly 1 | |
| Same rule says | 2³⁻³ = 2⁰ | so 2⁰ = 1 |
| Keep going down | 2⁻¹ = 1/2 | 2⁻³ = 1/8 |
| Negative exponent | means one over it | not negative |
A negative exponent is not a negative number
- Two to the minus three is one eighth. It is positive.
- The minus sign flips it over, it does not flip its sign.
- Keep halving past two to the zero and you land there.
Using several at once
- Three to the fifth times three to the minus two.
- Add the exponents: five and minus two make three.
- So it is three cubed, which is twenty-seven.
Roots: the other direction
- Squaring takes 3 to 9. A square root takes 9 back to 3.
- Most whole numbers are not perfect squares.
- Their roots are irrational, and that is the normal case.
Which roots are rational?
- Root nine is exactly 3, so it is rational.
- Root ten is 3.162 and running, so it is not.
- Same test for cube roots: perfect cube or not.
| Root | Value | Which is it |
|---|---|---|
| √9 | exactly 3 | rational |
| √10 | 3.16227766… | irrational |
| ³√27 | exactly 3 | rational |
| ³√10 | 2.15443469… | irrational |
Roots collect like terms
- Root three plus root three is two root threes.
- Same as two apples plus two apples being four apples.
- Root two plus root three will not collect. Different units.
| Expression | Simplifies to |
|---|---|
| √3 + √3 | 2√3 |
| 5√2 − 2√2 | 3√2 |
| √2 + √3 | stays exactly as it is |
| Same reason | 2x + 3y will not collect either |
Multiplying roots, and simplifying them
- Root two times root two is just two.
- Root two times root eight is root sixteen, which is 4.
- Pull perfect squares out: root eight is two root two.
| Expression | Becomes | Why |
|---|---|---|
| √2 × √2 | 2 | a root undoes a square |
| √2 × √8 | √16 = 4 | multiply inside |
| √8 | √4 × √2 | 2√2 |
| √50 | √25 × √2 | 5√2 |
Getting the root off the bottom
- One over root two has a root underneath. Convention says move it.
- Multiply top and bottom by root two, which is multiplying by 1.
- Nothing about the number changed. Only how it is written.
| Step | What it is | Effect |
|---|---|---|
| 1/√2 | a root underneath | untidy |
| Multiply by √2/√2 | which is 1 | nothing changes |
| Gives | √2 / 2 | root on top |
| 6/√3 | times √3/√3 | 2√3 |
Exact answers are the finished ones
- A circle of radius 3 has area nine pi.
- 28.27 is an approximation of that, not an improvement on it.
- Round once, at the end, and never in the middle.
| Answer | Exact | Rounded |
|---|---|---|
| Circle, radius 3 | area 9π | about 28.27 |
| Which is exact? | 9π is | 28.27 is not |
| √8 | 2√2 is exact | 2.83 is not |
| Round at the end | never in the middle |
What to take away
- Seven lines, and the first five are one idea.
| Multiplying powers | add the exponents |
| Dividing powers | subtract them |
| A power of a power | multiply them |
| Anything to the zero | is 1 |
| A negative exponent | means one over it |
| Roots collect like terms | √3 + √3 = 2√3 |
| An exact answer | keeps the root or the π |
Your turn: four questions
- Pause here and work all four.
- Question 2 wants both an answer and a simplification.
- Question 3 has an inside and an outside. Inside first.
| Question | |
|---|---|
| 1 | Simplify 3⁵ × 3⁻². Give a number. |
| 2 | Is √50 rational? Simplify it. |
| 3 | Work out (2⁴)² ÷ 2⁵. |
| 4 | Write 6/√3 with no root underneath. |
Show answer
1. Add the exponents: 5 and negative 2 make 3, so it is 3³, which is 27. 2. No. 50 is not a perfect square, so √50 is irrational. It does simplify, because 50 is 25 times 2 and 25 is a perfect square: √50 = 5√2, about 7.07. 3. The inside first: (2⁴)² is 2⁸. Then divide by 2⁵, subtracting exponents, giving 2³, which is 8. 4. Multiply top and bottom by √3, which is multiplying by 1 and changes nothing. That gives 6√3 over 3, which cancels to 2√3.
The answers
- Three to the fifth times three to the minus two is twenty-seven.
- Root fifty is irrational, and it simplifies to five root two.
- Two to the eighth over two to the fifth is two cubed, which is 8.
- Six over root three tidies to two root three.
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