Zero and Negative Indices

Worked example

1. 1

   $5^3÷5^3$ — Start with a number divided by itself. We will work it out two ways.

2. 2

   $5^3÷5^3=5^{3-3}$ — Way 1, the index law: when you divide powers of the same number, subtract the powers.

3. 3

   $=5^0$ — 3 − 3 = 0, so the answer is 5 to the power 0.

4. 4

   $5^3÷5^3=125÷125$ — Way 2, plain numbers: 5³ = 5 × 5 × 5 = 125.

5. 5

   $=1$ — Any number divided by itself is 1.

6. 6

   $5^0=1$ — Both ways started from the same sum, so their answers must be equal.

Worked example

1. 1

   $2^2=4$ — Start with a power you know: 2 × 2 = 4.

2. 2

   $2^1=2$ — One step down in the power means divide by 2: 4 ÷ 2 = 2.

3. 3

   $2^0=1$ — Divide by 2 again: 2 ÷ 2 = 1. This matches the last card.

4. 4

   $2^{-1}=\frac{1}{2}$ — Keep going past 0: 1 ÷ 2 = ½. Now the answers are fractions.

5. 5

   $2^{-2}=\frac{1}{4}$ — Divide by 2 again: ½ ÷ 2 = ¼.

6. 6

   $2^{-3}=\frac{1}{8}$ — Once more: ¼ ÷ 2 = ⅛.

7. 7

   $2^{-3}=\frac{1}{2^3}$ — Spot the pattern: 8 is 2³, so 2⁻³ is one over the positive power.

Worked example

1. 1

   $4^{-2}=\frac{1}{4^2}$ — The minus power means one over the positive power.

2. 2

   $=\frac{1}{16}$ — 4² = 4 × 4 = 16. Write this line down: it earns a mark even if the decimal goes wrong.

3. 3

   $=0.0625$ — 1 ÷ 16 = 0.0625. The question asked for a decimal, so finish here.

Worked example

1. 1

   $10^{-3}=\frac{1}{10^3}$ — The minus power means one over the positive power.

2. 2

   $=\frac{1}{1000}$ — 10³ = 10 × 10 × 10 = 1000.

3. 3

   $=0.001$ — One thousandth. The 1 is in the 3rd place after the point, matching the power 3.

Worked example

1. 1

   $0.00001$ — Count the places from the point to the 1. The 1 is in the 5th place.

2. 2

   $=\frac{1}{100000}$ — The 5th place after the point is hundred-thousandths.

3. 3

   $=\frac{1}{10^5}$ — 100000 is 1 with 5 zeros, so it is 10⁵.

4. 4

   $=10^{-5}$ — One over a power is the same power with a minus.

Worked example

1. 1

   ${\left(\frac{2}{5}\right)}^{-2}$ — The question. The minus in the power tells you to flip.

2. 2

   $={\left(\frac{5}{2}\right)}^2$ — Flip the fraction upside down. The power is now +2.

3. 3

   $=\frac{5^2}{2^2}$ — Square the top and the bottom separately.

4. 4

   $=\frac{25}{4}$ — 5² = 25 and 2² = 4. You can leave it as a top-heavy fraction or write 6¼.

Worked example

1. 1

   $\frac{5}{2^{-3}}$ — The question. A negative power sits on the bottom.

2. 2

   $2^{-3}=\frac{1}{8}$ — One over 2³, and 2³ = 8. Write this line: it earns a mark on its own.

3. 3

   $\frac{5}{2^{-3}}=5÷\frac{1}{8}$ — A fraction line means divide.

4. 4

   $=5\times 8$ — Dividing by ⅛ is the same as multiplying by 8, because 8 eighths make 1.

5. 5

   $=40$ — 5 × 8 = 40. A negative power on the bottom moves to the top as a positive power.

Worked example

1. 1

   $7^x=1$ — The question. Which power of 7 gives 1?

2. 2

   $x=0$ — Any number to the power 0 is 1, so the power must be 0.

Worked example

1. 1

   $3^x=\frac{1}{9}$ — The question. The answer is a fraction, so expect a minus power.

2. 2

   $9=3^2$ — Write the bottom as a power of 3: 3 × 3 = 9.

3. 3

   $\frac{1}{9}=\frac{1}{3^2}$ — Put that into the fraction.

4. 4

   $\frac{1}{3^2}=3^{-2}$ — One over a power is the same power with a minus.

5. 5

   $x=-2$ — Both sides are now powers of 3, so the powers match: x = −2.

Worked example

1. 1

   $2^x=0.125$ — The question. A decimal below 1, so again expect a minus power.

2. 2

   $0.125=\frac{1}{8}$ — Change the decimal to a fraction: 125 thousandths is ⅛.

3. 3

   $\frac{1}{8}=\frac{1}{2^3}$ — 8 = 2 × 2 × 2 = 2³.

4. 4

   $x=-3$ — One over 2³ is 2 to the power −3, so x = −3.

Worked example

1. 1

   $6^{-7}÷6^{-3}=6^n$ — The question. Same number, dividing, so subtract the powers.

2. 2

   $n=-7-(-3)$ — Take the second power from the first. Put −3 in brackets so its minus is not lost.

3. 3

   $n=-7+3$ — Taking away a negative is adding: −(−3) = +3.

4. 4

   $n=-4$ — −7 + 3 = −4.

Worked example

1. 1

   $\frac{7^8\times 7^{-5}}{7^{-2}}=7^k$ — The question. Do the top first, then the divide.

2. 2

   $7^8\times 7^{-5}=7^{8+(-5)}$ — Multiplying: add the powers. Keep −5 in brackets.

3. 3

   $=7^3$ — 8 + (−5) = 8 − 5 = 3. Write this line: it is the first mark.

4. 4

   $\frac{7^3}{7^{-2}}=7^{3-(-2)}$ — Dividing: take the bottom power from the top power.

5. 5

   $=7^{3+2}$ — Taking away −2 is adding 2.

6. 6

   $=7^5,\text{so}k=5$ — 3 + 2 = 5.

Worked example

1. 1

   $4^{-6}\times 4^6=4^{-6+6}$ — Multiplying: add the powers.

2. 2

   $=4^0$ — −6 + 6 = 0.

3. 3

   $=1$ — Any number to the power 0 is 1.