Contemporary Mathematics

# A | Co-Req Appendix: Integer Powers of 10

Contemporary MathematicsA | Co-Req Appendix: Integer Powers of 10

## Nonnegative Integer Powers of 10

The phrase nonnegative integers refers to the set containing 0, 1, 2, 3, … and so on. In the expression $105105$, 10 is called the base, and 5 is called the exponent, or power. The exponent 5 is telling us to multiply the base 10 by itself 5 times. So, $105=10×10×10×10×10=100,000105=10×10×10×10×10=100,000$. By definition, any number raised to the 0 power is 1. So, $100=1100=1$.

In the following table, there are several nonnegative integer powers of 10 that have been written as a product. Notice that higher exponents result in larger products. What do you notice about the number of zeros in the resulting product?

Exponential Form Product Number of Zeros in Product
$100100$ $11$ $00$
$101101$ $1010$ $11$
$102102$ $10×10=10010×10=100$ $22$
$103103$ $10×10×10=1,00010×10×10=1,000$ $33$
$104104$ $10×10×10×10=10,00010×10×10×10=10,000$ $44$
$105105$ $10×10×10×10×10=100,00010×10×10×10×10=100,000$ $55$

That’s right! The number of zeros is the same as the power each time!

## Negative Integer Powers of 10

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 10 is $110110$. We use negative exponents to indicate a reciprocal. For example, $10−1=1101=11010−1=1101=110$. Similarly, any expression with a negative exponent can be written with a positive exponent by taking the reciprocal. Several negative powers of 10 have been simplified in the table that follows. What do you notice about the number of zeros in the denominator (bottom) of each fraction?

Exponential Form Equivalent Simplified Expression Number of Zeros in Denominator
$10−110−1$ $1101=1101101=110$ $11$
$10−210−2$ $1102=110×10=11001102=110×10=1100$ $22$
$10−310−3$ $1103=110×10×10=11,0001103=110×10×10=11,000$ $33$
$10−410−4$ $1104=110×10×10×10=110,0001104=110×10×10×10=110,000$ $44$

That’s right! The number of zeros is the same as the positive version of the power each time.

In the following table, we will write the same powers of 10 as decimals. Count the number of decimal places to the right of the decimal point. What do you notice?

Exponential Form Equivalent Simplified Expression Number of Decimal Places to Right of Decimal
$10−110−1$ $1101=1÷10=0.11101=1÷10=0.1$ $11$
$10−210−2$ $1102=1÷100=0.011102=1÷100=0.01$ $22$
$10−310−3$ $1103=1÷1,000=0.0011103=1÷1,000=0.001$ $33$
$10−410−4$ $1104=1÷10,000=0.00011104=1÷10,000=0.0001$ $44$

That’s right! The number of decimal places to the right of the decimal point is the same as the positive version of the power each time.

## Multiplying Integers by Positive Powers of 10

Did you know that the distance from the sun to Earth is over 90 million miles? This value can be represented as 90,000,000, or we can write it as a product: $9×10,000,000=9×1079×10,000,000=9×107$, which is actually a more compact way of writing 90 million. Notice that the power of 7 reflects the number of zeros in 90 million. Several products of positive integers and powers of 10 are given in the table that follows. Notice that the number of zeros is the same as the exponent except in one case.

Exponential Form Product Number of Zeros in Product
$5×1015×101$ $5×10=505×10=50$ $11$
$13×10213×102$ $13×100=1,30013×100=1,300$ $22$
$8×1038×103$ $8×1,000=8,0008×1,000=8,000$ $33$
$15×10415×104$ $15×10,000=150,00015×10,000=150,000$ $44$
$70×10570×105$ $70×100,000=7,000,00070×100,000=7,000,000$ $66$

The only case in which the number of zeros didn’t equal the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70 which also had a zero. So, the product had a zero from the 70 and 5 zeros from $105105$ for a total of 6 zeros in 7,000,000.

## Multiplying by Negative Powers of 10

As we have seen, negative powers of 10 are decimals. Several products of positive integers and powers of 10 are given in the table below. Notice that multiplying an integer by 10 raised to a negative integer power results in a smaller number than you started with. Also, the number of decimal places to the right of the decimal point is the same as the exponent except in one case.

Exponential Form Product Number of Decimal Places to Right of Decimal
$3×10−13×10−1$ $3×0.1=0.33×0.1=0.3$ $11$
$13×10−213×10−2$ $13×0.01=0.1313×0.01=0.13$ $22$
$9×10−39×10−3$ $9×0.001=0.0099×0.001=0.009$ $33$
$15×10−415×10−4$ $15×0.0001=0.001515×0.0001=0.0015$ $44$
$70×10−570×10−5$ $70×0.00001=0.00070or0.000770×0.00001=0.00070or0.0007$ $5(6if we leave on the extra0)5(6if we leave on the extra0)$

The only case in which the number of decimal places to the right of the decimal point didn’t equal the positive version of the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70, which ended in zero.

## Moving the Decimal Place

A helpful shortcut when multiplying a number by a power of 10 is to “move the decimal point.” The following table shows several powers of 10, both positive and negative. Compare the location of the decimal point in the original number to the location of the decimal point in the product. How has it changed?

Exponential Form Product How the Position of the Decimal Point Changed
$5×1015×101$ $5.×10=5∧0⌣.=505.×10=5∧0⌣.=50$ $1place to the right1place to the right$
$13×10213×102$ $13.×100=130⌣∧0⌣.=1,30013.×100=130⌣∧0⌣.=1,300$ $2places to the right2places to the right$
$8×1038×103$ $8.×1,000=8∧0⌣0⌣0⌣.=8,0008.×1,000=8∧0⌣0⌣0⌣.=8,000$ $3places to the right3places to the right$
$15×10415×104$ $15.×10000=15∧0⌣0⌣0⌣0⌣.=150,00015.×10000=15∧0⌣0⌣0⌣0⌣.=150,000$ $4places to the right4places to the right$
$70×10570×105$ $70.×100,000=70∧0⌣0⌣0⌣0⌣0⌣.=7,000,00070.×100,000=70∧0⌣0⌣0⌣0⌣0⌣.=7,000,000$ $5places to the right5places to the right$
$3×10−13×10−1$ $3.×0.1=.3⌣∧=0.33.×0.1=.3⌣∧=0.3$ $1place to the left1place to the left$
$13×10−213×10−2$ $13.×0.01=.1⌣3⌣∧=0.1313.×0.01=.1⌣3⌣∧=0.13$ $2places to the left2places to the left$
$9×10−39×10−3$ $9.×0.001=.0⌣0⌣9⌣∧=0.0099.×0.001=.0⌣0⌣9⌣∧=0.009$ $3places to the left3places to the left$
$15×10−415×10−4$ $15.×0.0001=.0⌣0⌣1⌣5⌣∧=0.001515.×0.0001=.0⌣0⌣1⌣5⌣∧=0.0015$ $4places to the left4places to the left$
$70×10−570×10−5$ $70×0.00001=.0⌣0⌣0⌣7⌣0⌣∧=0.000770×0.00001=.0⌣0⌣0⌣7⌣0⌣∧=0.0007$ $5places to the left5places to the left$

Notice that multiplying by a positive power of 10 moves the decimal point to the right, making the value larger, while multiplying by a negative power of 10 moves the decimal point to the left, making the value smaller. Also, the number of decimal places that the decimal point moves is exactly the positive version of the exponent.