7th Grade Math · Unit 2
See how powers of 10 keep the same pattern when the exponent becomes negative.
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Down one exponent, divide by 10
Start at a power of 10 you know. Each time the exponent decreases by 1, divide the value by 10.
10² = 100, so 10¹ = 10, and 10⁰ = 1.
10 to the zero is 1
10⁰ = 1. That is the bridge from whole numbers into decimals.
The next step down divides 1 by 10.
A negative exponent is not a negative number
The next power after 10⁰ is 10⁻¹.
1 ÷ 10 = 0.1, so 10⁻¹ = 0.1. The value stays positive.
Keep going the same way
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01, so 10⁻² = 0.01.
Every new negative step is another division by 10. The value gets smaller, but it does not turn negative.
One step at a time. The exponent goes down by 1, and the value is divided by 10.
Each time the exponent decreases by 1, divide the value by 10.
The next exponent is 0 − 1 = −1.
1 ÷ 10 = 0.1
10⁻¹ = 0.1
The value is positive. A negative exponent does not make a negative number.
Answer: 10⁻¹ = 0.1
Each time the exponent decreases by 1, divide the value by 10.
100 ÷ 10 = 10
10¹ = 10
Answer: 10¹ = 10
Each time the exponent decreases by 1, divide the value by 10.
10,000 ÷ 10 = 1,000
10³ = 1,000
Answer: 10³ = 1,000
Each time the exponent decreases by 1, divide the value by 10.
100 ÷ 10 = 10
10¹ = 10
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
Answer: 10¹ = 10, 10⁰ = 1, 10⁻¹ = 0.1
Each value is the one before it divided by 10.
1 ÷ 10 = 0.1
100 is 10², 10 is 10¹, and 1 is 10⁰.
The next power is 10⁻¹.
Answer: 0.1, which is 10⁻¹
The list is 2, 1, 0, then the blank.
The next exponent is 0 − 1 = −1.
Start from 10⁰ = 1.
1 ÷ 10 = 0.1
0.1 is positive. The exponent is negative, and the value is not.
Answer: Exponent −1, and the value 0.1 is positive
Each time the exponent decreases by 1, divide the value by 10.
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
Answer: 10⁻¹ = 0.1
Each time the exponent decreases by 1, divide the value by 10.
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
Answer: 10⁰ = 1 and 10⁻¹ = 0.1
Follow the pattern several steps into the negative exponents.
Each value is the one before it divided by 10, so each exponent is 1 less.
10 = 10¹
1 = 10⁰
0.1 = 10⁻¹
The pattern crosses from whole numbers into a positive decimal.
Answer: 10¹, 10⁰, and 10⁻¹
Each time the exponent decreases by 1, divide the value by 10.
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001
The exponents decrease by 1: 2, then 1, then 0, then −1.
The missing exponent is 1.
100 ÷ 10 = 10, so 10¹ = 10.
Answer: 10¹ = 10
Each time the exponent decreases by 1, divide the value by 10.
1,000 ÷ 10 = 100
10² = 100
100 ÷ 10 = 10
10¹ = 10
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
That is 4 divisions.
Answer: 4 times, and 10⁻¹ = 0.1
1 = 10⁰
Divide by 10: 0.1 = 10⁻¹
Divide by 10 again: 0.01 = 10⁻²
Divide by 10 again: 0.001 = 10⁻³
Answer: 10⁰, 10⁻¹, 10⁻², 10⁻³
10⁻¹ = 0.1
One more step down divides by 10 again.
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 is smaller than 0.1. The more negative exponent is the smaller value.
Answer: 10⁻² = 0.01, which is smaller than 0.1
Each time the exponent decreases by 1, divide the value by 10.
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁰ = 1, 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001
| Power | Value |
|---|---|
| 10¹ | 10 |
| 10⁰ | 1 |
| 10⁻¹ | 0.1 |
| 10⁻² | ___ |
Each row divides the value above it by 10.
0.1 ÷ 10 = 0.01
| Power | Value |
|---|---|
| 10¹ | 10 |
| 10⁰ | 1 |
| 10⁻¹ | 0.1 |
| 10⁻² | 0.01 |
Answer: 10⁻² = 0.01
Each time the exponent decreases by 1, divide the value by 10.
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁻² = 0.01 and 10⁻³ = 0.001
Each time the exponent decreases by 1, divide the value by 10.
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁻³ = 0.001
Explain the pattern, and use it to reach smaller powers of 10.
Each time the exponent decreases by 1, divide the value by 10.
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 is greater than zero. The exponent changed sign. The value did not.
Answer: 10⁻¹ = 0.1, which is positive
Each time the exponent decreases by 1, divide the value by 10.
1,000 ÷ 10 = 100
10² = 100
100 ÷ 10 = 10
10¹ = 10
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁻³ = 0.001
Each time the exponent decreases by 1, divide the value by 10.
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
0.001 ÷ 10 = 0.0001
10⁻⁴ = 0.0001
Every step divides a positive number by 10.
The result stays positive. Only the exponent is negative.
Answer: 10⁻⁴ = 0.0001, which is positive
Each time the exponent decreases by 1, divide the value by 10.
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
0.001 ÷ 10 = 0.0001
10⁻⁴ = 0.0001
That is 4 divisions. The decimal point moved 4 places.
Answer: 4 times, and 10⁻⁴ = 0.0001
Each time the exponent decreases by 1, divide the value by 10.
0.0001 ÷ 10 = 0.00001
10⁻⁵ = 0.00001
0.00001 ÷ 10 = 0.000001
10⁻⁶ = 0.000001
Answer: 10⁻⁵ = 0.00001 and 10⁻⁶ = 0.000001
Each step down divides by 10, so the value gets smaller.
10⁰ = 1
10⁻¹ = 0.1
10⁻³ = 0.001
0.001 is smallest and 1 is largest.
10⁻³ < 10⁻¹ < 10⁰
Answer: 10⁻³ < 10⁻¹ < 10⁰
Each time the exponent decreases by 1, divide the value by 10.
100 ÷ 10 = 10
10¹ = 10
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
0.001 ÷ 10 = 0.0001
10⁻⁴ = 0.0001
0.0001 ÷ 10 = 0.00001
10⁻⁵ = 0.00001
That is 7 divisions. The exponent is 2 − 7 = −5.
Answer: 7 divisions, and the power is 10⁻⁵
| Power | Value |
|---|---|
| 10² | 100 |
| 10¹ | 10 |
| 10⁰ | 1 |
| 10⁻¹ | 0.1 |
| 10⁻² | 0.01 |
| 10⁻³ | 0.001 |
| 10⁻⁴ | 0.0001 |
Answer: From 10² = 100 down to 10⁻⁴ = 0.0001
Each time the exponent decreases by 1, divide the value by 10.
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
0.001 ÷ 10 = 0.0001
10⁻⁴ = 0.0001
0.0001 ÷ 10 = 0.00001
10⁻⁵ = 0.00001
Answer: 10⁻⁵ = 0.00001
Each time the exponent decreases by 1, divide the value by 10.
100,000 ÷ 10 = 10,000
10⁴ = 10,000
10,000 ÷ 10 = 1,000
10³ = 1,000
1,000 ÷ 10 = 100
10² = 100
100 ÷ 10 = 10
10¹ = 10
10 ÷ 10 = 1
10⁰ = 1
1 ÷ 10 = 0.1
10⁻¹ = 0.1
0.1 ÷ 10 = 0.01
10⁻² = 0.01
0.01 ÷ 10 = 0.001
10⁻³ = 0.001
Answer: 10⁵ down to 10⁻³ = 0.001
10 to a positive exponent is 1 followed by that many zeros.
10,000,000 is 1 followed by 7 zeros.
That is 10⁷.
The exponent is 7.
Answer: 7
10 is 1 followed by 1 zero.
That is 10¹.
The exponent is 1.
Answer: 1
100 is 1 followed by 2 zeros.
That is 10².
The exponent is 2.
Answer: 2
1,000 is 1 followed by 3 zeros.
That is 10³.
The exponent is 3.
Answer: 3
100,000 is 1 followed by 5 zeros.
That is 10⁵.
The exponent is 5.
Answer: 5
1,000,000 is 1 followed by 6 zeros.
That is 10⁶.
The exponent is 6.
Answer: 6
1,000,000,000 is 1 followed by 9 zeros.
That is 10⁹.
The exponent is 9.
Answer: 9
10 to a negative exponent is one tenth multiplied that many times.
A negative exponent counts factors of 110.
10⁻⁴ = 110⁴.
110 is multiplied 4 times.
Answer: 4 times
A negative exponent counts factors of 110.
10⁻² = 110².
110 is multiplied 2 times.
Answer: 2 times
A negative exponent counts factors of 110.
10⁻³ = 110³.
110 is multiplied 3 times.
Answer: 3 times
A negative exponent counts factors of 110.
10⁻⁵ = 110⁵.
110 is multiplied 5 times.
Answer: 5 times
A negative exponent counts factors of 110.
10⁻⁶ = 110⁶.
110 is multiplied 6 times.
Answer: 6 times
A negative exponent counts factors of 110.
10⁻⁷ = 110⁷.
110 is multiplied 7 times.
Answer: 7 times
A negative exponent counts factors of 110.
10⁻⁹ = 110⁹.
110 is multiplied 9 times.
Answer: 9 times
Match the decimal to a negative power of 10, then count the factors of one tenth.
0.001 = 10⁻³.
The exponent is −3, so 110 is a factor 3 times.
Answer: 3 times
0.1 = 10⁻¹.
The exponent is −1, so 110 is a factor 1 time.
Answer: 1 time
0.01 = 10⁻².
The exponent is −2, so 110 is a factor 2 times.
Answer: 2 times
0.00001 = 10⁻⁵.
The exponent is −5, so 110 is a factor 5 times.
Answer: 5 times
0.000001 = 10⁻⁶.
The exponent is −6, so 110 is a factor 6 times.
Answer: 6 times
0.0000001 = 10⁻⁷.
The exponent is −7, so 110 is a factor 7 times.
Answer: 7 times
0.000000001 = 10⁻⁹.
The exponent is −9, so 110 is a factor 9 times.
Answer: 9 times
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