7th Grade Math · Unit 2
Write a negative power of 10 as a fraction and as a decimal.
Negative exponent, reciprocal
10⁻¹ means the reciprocal of 10¹.
10⁻¹ = 110¹ = 110.
Then write the decimal
110 = 0.1.
1100 = 0.01.
11,000 = 0.001.
The exponent counts the decimal places
In 10⁻⁴ = 0.0001, the exponent is −4 and the 1 is four places after the decimal point.
The 1 sits in the ten-thousandths place. The value is positive.
Three forms, one number
10⁻³ = 11,000 = 0.001.
You can start from the power, the fraction, or the decimal. Write the other two.
Exponents −1, −2, and −3. Write the fraction and the decimal.
A negative exponent means the reciprocal. 10⁻¹ = 110¹.
10¹ = 10.
10⁻¹ = 110.
The 1 is in the tenths place, so the decimal has 1 place.
10⁻¹ = 0.1.
Answer: 110 and 0.1
A negative exponent means the reciprocal. 10⁻² = 110².
10² = 100.
10⁻² = 1100.
The 1 is in the hundredths place, so the decimal has 2 places.
10⁻² = 0.01.
Answer: 1100 and 0.01
A negative exponent means the reciprocal. 10⁻³ = 110³.
10³ = 1,000.
10⁻³ = 11,000.
The 1 is in the thousandths place, so the decimal has 3 places.
10⁻³ = 0.001.
Answer: 11,000 and 0.001
1100 = 110².
The denominator is 10 to the 2, so this is 10⁻².
10⁻² = 0.01.
Answer: 10⁻² and 0.01
11,000 = 110³.
The denominator is 10 to the 3, so this is 10⁻³.
10⁻³ = 0.001.
Answer: 10⁻³ and 0.001
In 0.01, the 1 is in the hundredths place.
That place is 2 digits after the decimal point, so the exponent is −2.
0.01 = 10⁻².
10⁻² = 1100.
Answer: 10⁻² and 1100
In 0.001, the 1 is in the thousandths place.
That place is 3 digits after the decimal point, so the exponent is −3.
0.001 = 10⁻³.
10⁻³ = 11,000.
Answer: 10⁻³ and 11,000
A negative exponent means the reciprocal. 10⁻² = 110².
10² = 100.
10⁻² = 1100.
The 1 is in the hundredths place, so the decimal has 2 places.
10⁻² = 0.01.
Answer: 10⁻² = 1100 = 0.01
A negative exponent means the reciprocal. 10⁻³ = 110³.
10³ = 1,000.
10⁻³ = 11,000.
The 1 is in the thousandths place, so the decimal has 3 places.
10⁻³ = 0.001.
0.0001 has the 1 in the ten-thousandths place. That is 10⁻⁴, not 10⁻³.
Answer: No. 10⁻³ = 0.001
110 = 110¹.
The denominator is 10 to the 1, so this is 10⁻¹.
10⁻¹ = 0.1.
Answer: 10⁻¹ and 0.1
In 0.1, the 1 is in the tenths place.
That place is 1 digit after the decimal point, so the exponent is −1.
0.1 = 10⁻¹.
10⁻¹ = 110.
Answer: 10⁻¹ and 110
Exponents −4, −5, and −6.
A negative exponent means the reciprocal. 10⁻⁴ = 110⁴.
10⁴ = 10,000.
10⁻⁴ = 110,000.
The 1 is in the ten-thousandths place, so the decimal has 4 places.
10⁻⁴ = 0.0001.
Answer: 110,000 and 0.0001
A negative exponent means the reciprocal. 10⁻⁵ = 110⁵.
10⁵ = 100,000.
10⁻⁵ = 1100,000.
The 1 is in the hundred-thousandths place, so the decimal has 5 places.
10⁻⁵ = 0.00001.
Answer: 1100,000 and 0.00001
A negative exponent means the reciprocal. 10⁻⁶ = 110⁶.
10⁶ = 1,000,000.
10⁻⁶ = 11,000,000.
The 1 is in the millionths place, so the decimal has 6 places.
10⁻⁶ = 0.000001.
Answer: 11,000,000 and 0.000001
110,000 = 110⁴.
The denominator is 10 to the 4, so this is 10⁻⁴.
10⁻⁴ = 0.0001.
Answer: 10⁻⁴ and 0.0001
1100,000 = 110⁵.
The denominator is 10 to the 5, so this is 10⁻⁵.
10⁻⁵ = 0.00001.
Answer: 10⁻⁵ and 0.00001
11,000,000 = 110⁶.
The denominator is 10 to the 6, so this is 10⁻⁶.
10⁻⁶ = 0.000001.
Answer: 10⁻⁶ and 0.000001
In 0.0001, the 1 is in the ten-thousandths place.
That place is 4 digits after the decimal point, so the exponent is −4.
0.0001 = 10⁻⁴.
10⁻⁴ = 110,000.
Answer: 10⁻⁴ and 110,000
In 0.00001, the 1 is in the hundred-thousandths place.
That place is 5 digits after the decimal point, so the exponent is −5.
0.00001 = 10⁻⁵.
10⁻⁵ = 1100,000.
Answer: 10⁻⁵ and 1100,000
In 0.000001, the 1 is in the millionths place.
That place is 6 digits after the decimal point, so the exponent is −6.
0.000001 = 10⁻⁶.
10⁻⁶ = 11,000,000.
Answer: 10⁻⁶ and 11,000,000
A negative exponent means the reciprocal. 10⁻⁶ = 110⁶.
10⁶ = 1,000,000.
10⁻⁶ = 11,000,000.
The 1 is in the millionths place, so the decimal has 6 places.
10⁻⁶ = 0.000001.
The positive version of the exponent tells the number of decimal places.
Answer: 6 digits, because the exponent is −6
A negative exponent means the reciprocal. 10⁻⁵ = 110⁵.
10⁵ = 100,000.
10⁻⁵ = 1100,000.
The 1 is in the hundred-thousandths place, so the decimal has 5 places.
10⁻⁵ = 0.00001.
Answer: 10⁻⁵ = 1100,000 = 0.00001
Correct a mistake, or compare a fraction and a decimal.
The reciprocal of 10³ is 1 divided by 10³, not 1 divided by the decimal.
10³ = 1,000.
10⁻³ = 11,000 = 0.001.
1 divided by 0.001 is 1,000, which is 10³, the opposite of 10⁻³.
Answer: 10⁻³ = 11,000 = 0.001
A negative exponent does not put a negative sign on the value.
A negative exponent means the reciprocal. 10⁻² = 110².
10² = 100.
10⁻² = 1100.
The 1 is in the hundredths place, so the decimal has 2 places.
10⁻² = 0.01.
Answer: 10⁻² = 1100 = 0.01
A negative exponent means the reciprocal. 10⁻⁴ = 110⁴.
10⁴ = 10,000.
10⁻⁴ = 110,000.
The 1 is in the ten-thousandths place, so the decimal has 4 places.
10⁻⁴ = 0.0001.
0.00001 is 10⁻⁵, one more place than 10⁻⁴.
Answer: 0.0001
A negative exponent means the reciprocal. 10⁻⁵ = 110⁵.
10⁵ = 100,000.
10⁻⁵ = 1100,000.
The 1 is in the hundred-thousandths place, so the decimal has 5 places.
10⁻⁵ = 0.00001.
110,000 is 10⁻⁴, not 10⁻⁵.
Answer: 1100,000
110,000 = 110⁴.
The denominator is 10 to the 4, so this is 10⁻⁴.
10⁻⁴ = 0.0001.
In 0.00001, the 1 is in the hundred-thousandths place.
That place is 5 digits after the decimal point, so the exponent is −5.
0.00001 = 10⁻⁵.
10⁻⁵ = 1100,000.
10⁻⁴ is one step above 10⁻⁵, so it is greater.
110,000 > 0.00001
Answer: 110,000 > 0.00001
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