Zero and Negative Indices with Letters

Worked example

1. 1

   $a^5÷a^5$ — Start with something divided by itself. Work it out two ways.

2. 2

   $=a^{5-5}$ — Way 1: dividing powers of the same letter, so take away the powers.

3. 3

   $=a^0$ — 5 − 5 = 0, so the answer is a to the power 0.

4. 4

   $a^5÷a^5=1$ — Way 2: anything (except 0) divided by itself is 1.

5. 5

   $a^0=1$ — Both ways start from the same thing, so they must be equal.

Worked example

1. 1

   $6x^0=6\times x^0$ — Write the hidden ×. The 0 sits on the x only, not on the 6.

2. 2

   $=6\times 1$ — x to the power 0 is 1, like every power 0.

3. 3

   $=6$ — 6 × 1 = 6. The 6 was never powered, so it stays.

4. 4

   $3(2k{)}^0=3\times (2k{)}^0$ — Now a bracket. The 0 is on the whole bracket, but not on the 3 in front.

5. 5

   $=3\times 1=3$ — The bracket becomes 1, and 3 × 1 = 3.

6. 6

   $\frac{4}{w^0}=\frac{4}{1}$ — The 0 is on the w only, so the bottom becomes 1.

7. 7

   $=4$ — Dividing by 1 changes nothing, so the answer is 4.

Worked example

1. 1

   $2^3=8$ — Start with a power you know: 2 × 2 × 2 = 8.

2. 2

   $2^2=4,2^1=2$ — Each time the power goes down by 1, divide by 2 once more.

3. 3

   $2^0=1$ — One more step down: 2 ÷ 2 = 1. This matches the power 0 rule.

4. 4

   $2^{-1}=\frac{1}{2}$ — Keep going: 1 ÷ 2 = ½. The power is now negative, and the answer is a fraction.

5. 5

   $2^{-2}=\frac{1}{4}$ — ½ ÷ 2 = ¼, which is one over 2².

6. 6

   $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$ — ¼ ÷ 2 = ⅛. It is one over the positive power 2³.

Worked example

1. 1

   $3x^{-2}=3\times x^{-2}$ — Write the hidden ×. The −2 is on the x only, not on the 3.

2. 2

   $=3\times \frac{1}{x^2}$ — A negative power means one over, so $x^{-2}$ is one over x².

3. 3

   $=\frac{3}{x^2}$ — 3 times one over x² is 3 over x². The 3 stays on top, because it was never powered.

Worked example

1. 1

   $\frac{5}{x^4}=5\times \frac{1}{x^4}$ — Now the other way, into the form $p{x}^n$. First split off the 5.

2. 2

   $=5\times x^{-4}$ — One over x⁴ is x to the −4, by the formula read backwards.

3. 3

   $=5x^{-4}$ — Write it as one term. So p = 5 and n = −4.

Worked example

1. 1

   $\frac{4}{y^{-2}}=4÷y^{-2}$ — A negative power on the bottom. The fraction line means divide.

2. 2

   $=4÷\frac{1}{y^2}$ — y⁻² is one over y², by the formula.

3. 3

   $=4\times y^2$ — Dividing by a fraction is the same as multiplying by it flipped.

4. 4

   $=4y^2$ — So a negative power on the bottom moves to the top as a positive power.

Worked example

1. 1

   $y^8\times y^{-2}$ — Same letter, multiplied, so the powers will be added.

2. 2

   $=y^{8+(-2)}$ — Add the powers. Keep −2 in brackets so its minus stays with it.

3. 3

   $=y^6$ — 8 + (−2) = 8 − 2 = 6. Adding a negative is the same as taking away.

Worked example

1. 1

   $x^4÷x^{-6}$ — Same letter, divided, so the powers will be subtracted.

2. 2

   $=x^{4-(-6)}$ — Take away the second power. The brackets keep −6 whole.

3. 3

   $=x^{4+6}$ — Taking away a negative is the same as adding: −(−6) = +6.

4. 4

   $=x^{10}$ — 4 + 6 = 10.

Worked example

1. 1

   $x^5÷x^k=x^{-2}$ — Both sides are powers of x, so their powers must be equal.

2. 2

   $5-k=-2$ — Dividing takes away the powers, so the left side is x to the 5 − k.

3. 3

   $-k=-2-5$ — Take 5 from both sides to get k on its own.

4. 4

   $-k=-7$ — −2 − 5 = −7.

5. 5

   $k=7$ — Multiply both sides by −1. Check: 5 − 7 = −2, which is right.

Worked example

1. 1

   ${\left(\frac{3x^2}{12x^{5}y}\right)}^{-2}$ — The question. Do not use the power yet: tidy up inside the bracket first.

2. 2

   $={\left(\frac{x^2}{4x^{5}y}\right)}^{-2}$ — Cancel the numbers: 3 ÷ 3 = 1 on top, and 12 ÷ 3 = 4 on the bottom.

3. 3

   $={\left(\frac{1}{4x^{3}y}\right)}^{-2}$ — Cancel the x's: x⁵ ÷ x² = x³. It stays on the bottom, where the bigger power was.

4. 4

   $={\left(\frac{4x^{3}y}{1}\right)}^2$ — Flip the fraction, so the minus power becomes a plus power.

5. 5

   $=(4x^{3}y{)}^2$ — Anything over 1 is just itself, so drop the 1.

6. 6

   $=4^2\times x^{3\times 2}\times y^2$ — The power 2 goes on every part: the 4, the x³ and the y.

7. 7

   $=16x^6{y}^2$ — 4² = 16, and for x³ squared multiply the powers: 3 × 2 = 6.

Worked example

1. 1

   $(2m^3{)}^{-4}$ — The question. The −4 is on the whole bracket: the 2 and the m³.

2. 2

   $=\frac{1}{(2m^3{)}^4}$ — A negative power means one over. The minus goes, and the power becomes 4.

3. 3

   $=\frac{1}{2^4\times (m^3{)}^4}$ — The power 4 goes on every part inside the bracket: the number and the letter.

4. 4

   $=\frac{1}{16\times m^{12}}$ — 2⁴ = 2 × 2 × 2 × 2 = 16. For (m³)⁴ multiply the powers: 3 × 4 = 12.

5. 5

   $=\frac{1}{16m^{12}}$ — Write it as one fraction. This is the simplified answer.

6. 6

   $=\frac{1}{16}{m}^{-12}$ — Only if asked for the form $p{m}^n$: m¹² moves to the top and its power turns negative.