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Math 8 · 8.EE.A.4

Calculating in scientific notation

Two numbers are each written as a small number times a power of ten. How do you multiply, divide, add or subtract them without writing out all the zeros?

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Writing and reading scientific notationMath 8
Where you are 3 in Scientific notation
Number9
Exponents and roots5
Scientific notation3
Writing and reading scientific notation
Calculating in scientific notation
Comparing numbers in scientific notation

When you reach for it

A number in Scientific notation is a leading part between 1 and 10 times a power of ten, such as 2.5 \times 10^{13} or 7.5 \times 10^{-6}. Writing a number this way makes it short; this lesson is about calculating with such numbers.

Reach for it whenever a calculation mixes very large or very small numbers: how far light travels in a year, how many red blood cells are in a body, how many photos fit on a hard drive. Written out, 300\,000 \times 31\,500\,000 is a long multiplication with many zeros to count. In scientific notation it is 3 \times 3.15 and 5 + 7, two small calculations.

The work splits into two halves resting on two different facts.

  • Multiplying and dividing use the rules for powers with the same base: powers of ten multiply by adding exponents and divide by subtracting them. Leading parts and powers of ten are handled separately.
  • Adding and subtracting use the distributive property, which needs both numbers to have the same power of ten. So the first job is to make the powers match.

Deciding which of two numbers is bigger is its own lesson.

The procedure

The lesson is taught over two days. Part 1 is multiplying and dividing. Part 2 is adding and subtracting, choosing a sensible unit, and reading a calculator.

Part 1: multiplying and dividing

Multiplying. For any leading parts a and b and any integer exponents m and n,

\begin{aligned} (a \times 10^{m})(b \times 10^{n}) &= (a \times b) \times 10^{m+n} \end{aligned}

The left side is four numbers multiplied together. Multiplication may be reordered and regrouped (the commutative and associative properties of multiplication), so the leading parts can be put together and the powers of ten put together. The leading parts multiply as ordinary decimals. The powers combine by the product of powers rule, 10^{m} \times 10^{n} = 10^{m+n}: m factors of ten followed by n more is m + n factors of ten. The rule holds just as well when an exponent is zero or negative.

For instance, (2 \times 10^{3})(4 \times 10^{2}) = 8 \times 10^{5}, and in full that says 2000 \times 400 = 800\,000.

Putting the leading part back in range. The product of the leading parts can be 10 or more. Then the answer has the right value but is not yet in scientific notation:

(5 \times 10^{3})(4 \times 10^{6}) = 20 \times 10^{9}

Write 20 as 2 \times 10 and let that extra ten join the power:

\begin{aligned} 20 \times 10^{9} &= 2 \times 10 \times 10^{9} \\ &= 2 \times 10^{10} \end{aligned}

The leading part was divided by 10 and the exponent went up by 1; dividing and multiplying by 10 cancel, so the value did not change. Putting a leading part back between 1 and 10 this way is called renormalizing.

45
\times
10^{12}
4.5
\times
10^{13}
\div 10
\times 10
 after multiplying
5.9
\times
10^{5}
0.59
\times
10^{6}
\div 10
\times 10
 before adding
Figure 1One move does both jobs. The leading part gives up a factor of 10 and the power of ten takes it, so the value never changes. On the left it puts a product back in range; on the right it lines up two powers before adding.

After a multiplication this move is needed at most once. Each leading part is at least 1 and less than 10, so their product is at least 1 and less than 100: either it is already below 10, or one shift brings it there.

Dividing. For the same a, b, m and n,

\frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n}

Dividing by a product is dividing by each factor in turn, so the fraction splits into the leading parts and the powers. The powers combine by the quotient of powers rule, \displaystyle \frac{10^{m}}{10^{n}} = 10^{m-n}: the n factors of ten underneath cancel n of those on top. When n is negative the subtraction turns into an addition, so 10^{9} \div 10^{-3} = 10^{9 - (-3)} = 10^{12}. Dividing by a small number makes a result bigger, which is what the larger exponent says.

Renormalizing after a division. The quotient of two leading parts is more than 0.1 and less than 10, so it can fall below 1. Then a ten moves the other way, out of the power and into the leading part:

\begin{aligned} 0.4 \times 10^{7} &= 4 \times 10^{-1} \times 10^{7} \\ &= 4 \times 10^{6} \end{aligned}

The leading part was multiplied by 10 and the exponent went down by 1. Again one shift is always enough.

A quick sense check on the exponent: a number near 10^{5} times a number near 10^{8} must be near 10^{13} or 10^{14}, nowhere near 10^{40}.

Part 2: adding and subtracting

When the powers match. For any leading parts a and b,

\begin{aligned} a \times 10^{n} + b \times 10^{n} &= (a + b) \times 10^{n} \end{aligned}

This is the distributive property read from right to left: 10^{n} is a factor of both terms, so it can be taken out once, just as 3 thousands and 4 thousands make 7 thousands. For instance, 2.1 \times 10^{4} + 5.3 \times 10^{4} = 7.4 \times 10^{4}, which in full is 21\,000 + 53\,000 = 74\,000. Subtraction works the same way with a - b.

When the powers differ, make them match first. Use the move from Part 1, this time before adding. To add 3.2 \times 10^{5} and 6 \times 10^{4}, rewrite the second number with the power 10^{5}: 6 \times 10^{4} = 0.6 \times 10^{5}. The leading part was divided by 10 and the exponent went up by 1, so the value is the same. Now

\begin{aligned} 3.2 \times 10^{5} + 0.6 \times 10^{5} &= 3.8 \times 10^{5} \end{aligned}

In full, 320\,000 + 60\,000 = 380\,000. Rewriting the larger number to the smaller power, as 32 \times 10^{4}, gives the same value; matching the larger power is usually tidier because the answer needs little renormalizing. If the exponents differ by 3, the leading part moves three places: 5 \times 10^{2} = 0.005 \times 10^{5}.

Renormalize at the end. A sum of two leading parts can reach 10 or more, and one shift fixes it. A difference can fall below 1, and here, unlike everywhere else in this lesson, more than one shift may be needed:

\begin{aligned} 4.37 \times 10^{5} - 4.36 \times 10^{5} &= 0.01 \times 10^{5} \\ &= 1 \times 10^{3} \end{aligned}

Choosing a sensible unit. An answer such as 8.22 \times 10^{10} cubic meters is exact but hard to picture. A different unit can say the same amount with a number a reader can hold in mind. Metric units step by powers of ten (see Metric prefixes): kilo is 10^{3}, mega 10^{6}, giga 10^{9}, tera 10^{12}, milli 10^{-3}, micro 10^{-6}, nano 10^{-9}. Changing to a bigger unit is a division, exactly the Part 1 calculation. With a gigabyte as 10^{9} bytes, 3.6 \times 10^{10} bytes is

\begin{aligned} \frac{3.6 \times 10^{10}}{10^{9}} &= 3.6 \times 10^{1} \\ &= 36 \text{ gigabytes} \end{aligned}

Reading a calculator. When an answer is too long for the screen, a calculator switches to scientific notation, often written with the letter E: 4.5\text{E}13 means 4.5 \times 10^{13}, and 2.1\text{E-}7 means 2.1 \times 10^{-7}. The E is short for “exponent” and has nothing to do with the number e \approx 2.718.

To enter such a number, use the key marked EE or EXP: typing 3, EE, 5 enters 3 \times 10^{5} as one number. Typing the multiplication sign and the power by hand makes separate numbers, and the calculator then follows the order of operations, doing multiplications and divisions from left to right. So 8 \times 10^{12} \div 2 \times 10^{3} typed just like that means \left((8 \times 10^{12}) \div 2\right) \times 10^{3}, and the 10^{3} multiplies. Brackets around each number keep it in one piece. Example 7 works this through.

A calculator also shows more digits than the data supports; Significant figures says how far to round.

Worked examples

Examples 1 to 4 belong to Part 1, and Examples 5 to 7 to Part 2.

Example 1

Multiplying and dividing

Calculate, giving each answer in scientific notation.

(a) (2.5 \times 10^{6})(3 \times 10^{-2}) \qquad (b) (8.4 \times 10^{9}) \div (2 \times 10^{-3})

Solution

(a) Reorder and regroup so the leading parts sit together and the powers sit together. The leading parts give 2.5 \times 3 = 7.5, and the exponents add, 6 + (-2) = 4:

\begin{aligned} (2.5 \times 10^{6})(3 \times 10^{-2}) &= (2.5 \times 3) \times (10^{6} \times 10^{-2}) \\ &= 7.5 \times 10^{4} \end{aligned}

The leading part 7.5 is in range. Check in full: 2\,500\,000 \times 0.03 = 75\,000.

(b) The leading parts give 8.4 \div 2 = 4.2. The exponents subtract, and subtracting -3 is adding 3: 9 - (-3) = 12. So the answer is 4.2 \times 10^{12}.

The exponent went up because dividing by 2 \times 10^{-3} is dividing by 0.002, a small number. Check by multiplying back: (4.2 \times 10^{12})(2 \times 10^{-3}) = 8.4 \times 10^{9}.

Example 2

When the leading part leaves the range

Calculate, giving each answer in scientific notation.

(a) (6 \times 10^{4})(7.5 \times 10^{8}) \qquad (b) (1.2 \times 10^{5}) \div (4 \times 10^{-6})

Solution

(a) The leading parts give 6 \times 7.5 = 45 and the exponents give 4 + 8 = 12, so the product is 45 \times 10^{12}. The value is right, but 45 is not less than 10. Write 45 = 4.5 \times 10 and pass the ten to the power:

\begin{aligned} 45 \times 10^{12} &= 4.5 \times 10 \times 10^{12} \\ &= 4.5 \times 10^{13} \end{aligned}

This is the move in the left half of the figure.

(b) The leading parts give 1.2 \div 4 = 0.3 and the exponents give 5 - (-6) = 11, so the quotient is 0.3 \times 10^{11}. Now the leading part is too small. Write 0.3 = 3 \times 10^{-1}:

\begin{aligned} 0.3 \times 10^{11} &= 3 \times 10^{-1} \times 10^{11} \\ &= 3 \times 10^{10} \end{aligned}

Check by multiplying back: (3 \times 10^{10})(4 \times 10^{-6}) = 12 \times 10^{4} = 1.2 \times 10^{5}, the number we started with.

Example 3

How far light travels in a year

Light travels about 3 \times 10^{5} kilometers each second, and a year has about 3.15 \times 10^{7} seconds.

(a) About how far does light travel in one year? This distance is called a light-year.

(b) A distance of 3.78 \times 10^{13} kilometers is how many light-years?

Solution

(a) Distance is speed times time. The leading parts give 3 \times 3.15 = 9.45 and the exponents give 5 + 7 = 12:

\begin{aligned} (3 \times 10^{5})(3.15 \times 10^{7}) &= 9.45 \times 10^{12} \text{ km} \end{aligned}

Both inputs were rounded, so the last digit is not to be trusted: with more precise values the light-year is about 9.46 \times 10^{12} km.

(b) How many light-years fit into the distance is a division:

\begin{aligned} \frac{3.78 \times 10^{13}}{9.45 \times 10^{12}} &= \frac{3.78}{9.45} \times 10^{13 - 12} \\ &= 0.4 \times 10^{1} = 4 \end{aligned}

So the distance is 4 light-years: a small, plain number, which is why astronomers use the unit.

Example 4

Red blood cells in a body

Blood holds about 5 \times 10^{6} red blood cells in each cubic millimeter, and an adult has about 5 liters of blood.

(a) About how many red blood cells are in an adult’s blood?

(b) Each red blood cell is about 7.5 \times 10^{-6} meters across. If all of them were laid side by side in a line, about how long would the line be?

Solution

(a) First put the blood in the unit of the count. A liter is 10^{3} cubic centimeters and a cubic centimeter is 10^{3} cubic millimeters, so a liter is 10^{3} \times 10^{3} = 10^{6} cubic millimeters and five liters is 5 \times 10^{6} cubic millimeters. Cells per cubic millimeter times cubic millimeters:

\begin{aligned} (5 \times 10^{6})(5 \times 10^{6}) &= 25 \times 10^{12} \\ &= 2.5 \times 10^{13} \end{aligned}

That is about twenty-five million million red blood cells.

(b) Length is the number of cells times the width of each:

\begin{aligned} (2.5 \times 10^{13})(7.5 \times 10^{-6}) &= 18.75 \times 10^{7} \\ &= 1.875 \times 10^{8} \text{ m} \end{aligned}

The exponents gave 13 + (-6) = 7, and renormalizing 18.75 raised it to 8. In kilometers, divide by 10^{3}: 1.875 \times 10^{5} km. The inputs were rough, so “about 1.9 \times 10^{5} km”, nearly two hundred thousand kilometers, is the honest answer.

Example 5

Adding and subtracting

Calculate, giving each answer in scientific notation.

(a) 8.5 \times 10^{4} + 3.7 \times 10^{4} \qquad (b) 7.4 \times 10^{6} + 5.9 \times 10^{5} \qquad (c) 3.2 \times 10^{-4} - 8 \times 10^{-5}

Solution

(a) The powers already match, so add the leading parts and keep 10^{4}, by the distributive property:

\begin{aligned} (8.5 + 3.7) \times 10^{4} &= 12.2 \times 10^{4} \\ &= 1.22 \times 10^{5} \end{aligned}

(b) The powers differ. Rewrite the number with the smaller power: 5.9 \times 10^{5} = 0.59 \times 10^{6}, the move in the right half of the figure. Then

\begin{aligned} 7.4 \times 10^{6} + 0.59 \times 10^{6} &= 7.99 \times 10^{6} \end{aligned}

Check in full: 7\,400\,000 + 590\,000 = 7\,990\,000.

(c) The larger power here is 10^{-4}, since -4 is greater than -5. Rewrite 8 \times 10^{-5} = 0.8 \times 10^{-4}:

\begin{aligned} 3.2 \times 10^{-4} - 0.8 \times 10^{-4} &= 2.4 \times 10^{-4} \end{aligned}

Check in full: 0.000\,32 - 0.000\,08 = 0.000\,24.

Example 6

A country’s water use

In one year a country’s farms use 6.8 \times 10^{10} cubic meters of water, its industry uses 9.5 \times 10^{9} cubic meters and its homes use 4.7 \times 10^{9} cubic meters. (The country and its numbers are made up.)

(a) What is the country’s total water use for the year?

(b) How much more do the farms use than industry and homes together?

(c) Give the total in cubic kilometers.

Solution

(a) The largest power is 10^{10}, so rewrite the other two to match: 9.5 \times 10^{9} = 0.95 \times 10^{10} and 4.7 \times 10^{9} = 0.47 \times 10^{10}. Then

\begin{aligned} (6.8 + 0.95 + 0.47) \times 10^{10} &= 8.22 \times 10^{10} \text{ m}^3 \end{aligned}

(b) Industry and homes share the power 10^{9}, so add them first:

\begin{aligned} (9.5 + 4.7) \times 10^{9} &= 14.2 \times 10^{9} \\ &= 1.42 \times 10^{10} \text{ m}^3 \end{aligned}

Renormalizing has put this on the same power as the farms. Subtract:

\begin{aligned} 6.8 \times 10^{10} - 1.42 \times 10^{10} &= 5.38 \times 10^{10} \text{ m}^3 \end{aligned}

(c) A kilometer is 10^{3} meters, so a cubic kilometer is (10^{3})^{3} = 10^{9} cubic meters. Changing to the bigger unit divides by 10^{9}:

\begin{aligned} \frac{8.22 \times 10^{10}}{10^{9}} &= 8.22 \times 10^{1} \\ &= 82.2 \text{ km}^3 \end{aligned}
Example 7

Photos on a hard drive, and how a calculator reads a division

A hard drive holds 2 terabytes. Counted the way drive makers count, a terabyte is 10^{12} bytes and a megabyte is 10^{6} bytes. A photo takes about 4 megabytes.

(a) About how many photos fit on the drive?

(b) The same division is typed into a calculator as 2 \times 10^{12} \div 4 \times 10^{6}, with the power key and no brackets, and the screen shows 5\text{E}17. Using the order of operations, explain what the calculator computed, and how the entry must change to divide by the whole of 4 \times 10^{6}.

Solution

(a) The number of photos is the quotient of bytes on the drive by bytes per photo:

\begin{aligned} \frac{2 \times 10^{12}}{4 \times 10^{6}} &= 0.5 \times 10^{6} \\ &= 5 \times 10^{5} \end{aligned}

The leading parts gave 0.5, so one ten came out of the power. About 500\,000 photos fit.

(b) Typed this way, 2 \times 10^{12}, 4 and 10^{6} are separate numbers joined by \div and \times. Multiplication and division have equal rank and are carried out from left to right, so the calculator divides by 4 first and then multiplies the result by 10^{6}:

\begin{aligned} \left((2 \times 10^{12}) \div 4\right) \times 10^{6} &= (5 \times 10^{11}) \times 10^{6} \\ &= 5 \times 10^{17} \end{aligned}

The 10^{6} ended up as a factor of the top, not the bottom. Brackets, 2 \times 10^{12} \div (4 \times 10^{6}), or entering the divisor as one number with the EE key, 4\text{E}6, make the whole of 4 \times 10^{6} the divisor and give 500000. The sense check from Part 1 agrees: about 10^{12} divided by about 10^{6} is near 10^{6}.

Boundaries

The leading part must end between 1 and 10. The rules always give a correct value, but not always one in scientific notation. A product such as 45 \times 10^{12} or a quotient such as 0.3 \times 10^{11} is a true answer in the wrong form until it is renormalized.

Exponents combine only in products and quotients. In a product, 10^{m} \times 10^{n} = 10^{m+n}. A sum has no such rule: 10^{4} + 10^{4} = 2 \times 10^{4}, not 10^{8}. Adding and subtracting need the same power of ten, because the distributive property can only take out a factor the two terms share.

Zero has no scientific notation. No number between 1 and 10 times a power of ten equals 0, so a difference such as 4.2 \times 10^{6} - 4.2 \times 10^{6} is simply 0. And since a leading part is never 0, dividing by a number in scientific notation never divides by zero.

Negative numbers carry their sign in front. -3 \times 10^{4} is -30\,000. The sign follows the usual rules for signed numbers, and the powers of ten are handled exactly as above.

A much smaller number may vanish into the rounding. When the exponents differ by more than the number of digits kept, adding the smaller number does not change the digits shown. 6 \times 10^{9} + 2 \times 10^{1} is exactly 6.000\,000\,02 \times 10^{9}, which to two or three digits is still 6 \times 10^{9}. An answer is only as precise as its inputs, however many digits a calculator shows.

Storage is counted two ways. Drive makers use 10^{9} bytes for a gigabyte, while some computer software uses 2^{30} = 1\,073\,741\,824 bytes, about 7\% more. This lesson uses powers of ten throughout.

Where this goes next

Rounding a calculated leading part is the business of Significant figures. Chains of rough multiplications like Examples 3 and 4 are the method of estimating a huge quantity from small facts, where only the power of ten really matters. Every metric conversion is a multiplication or division by a power of ten, which is Metric prefixes put to work.

Multiplying numbers by adding exponents is also the idea behind a logarithm, which to base ten is the exponent itself. The same rules for any base are the Laws of exponents, and Exponential growth and decay is where scientific notation is used most.

Exercises

Part 1: multiplying

Give each answer in scientific notation.

  1. (2 \times 10^{3})(4 \times 10^{5})
  2. (1.5 \times 10^{-4})(6 \times 10^{9})
  3. (5 \times 10^{7})(8 \times 10^{-3})
  4. (9.2 \times 10^{-6})(5 \times 10^{-8})

Part 1: dividing

Give each answer in scientific notation.

  1. (6.3 \times 10^{11}) \div (3 \times 10^{4})
  2. (4.5 \times 10^{-2}) \div (9 \times 10^{5})
  3. (2.4 \times 10^{3}) \div (8 \times 10^{-6})
  4. \dfrac{(3 \times 10^{5})(8 \times 10^{-2})}{6 \times 10^{7}}

Part 1: problems

  1. A computer carries out about 2.5 \times 10^{9} operations each second. About how long does it take to carry out 10^{12} operations? Give the answer in seconds and in minutes.
  2. Light travels about 3 \times 10^{5} kilometers each second, and an hour has 3.6 \times 10^{3} seconds. How far does light travel in an hour?
  3. Ines-Maria’s blood sample has 4.8 \times 10^{6} red blood cells in each cubic millimeter. How many red blood cells are in 2.5 \times 10^{-2} cubic millimeters of it?
  4. Find the number, in scientific notation, that makes (4 \times 10^{6}) \times \underline{\quad} = 2 \times 10^{-3} true.

Part 2: adding and subtracting

Give each answer in scientific notation.

  1. 3.6 \times 10^{9} + 5.1 \times 10^{9}
  2. 4.25 \times 10^{6} + 7 \times 10^{4}
  3. 9.1 \times 10^{-3} - 6 \times 10^{-4}
  4. 7.8 \times 10^{5} + 4.6 \times 10^{5}
  5. 2.05 \times 10^{8} - 1.8 \times 10^{8}

Part 2: units and calculators

  1. In a made-up country, homes use 3.4 \times 10^{9} cubic meters of water a year and all other uses together take 2.66 \times 10^{10} cubic meters. Find the total in cubic meters, then in cubic kilometers (1 \text{ km}^3 = 10^{9} \text{ m}^3).
  2. Jonte’s hard drive holds 4 terabytes, and 7.5 \times 10^{11} bytes are in use. Taking a terabyte as 10^{12} bytes and a gigabyte as 10^{9} bytes, how much space is free, in bytes and in gigabytes?
  3. A calculator shows 3.7\text{E-}6 as the answer to a length in meters. Write the length in scientific notation and as a decimal, then in micrometers, where a micrometer is 10^{-6} meters.
  4. A star is 1.89 \times 10^{14} kilometers away. Taking a light-year as 9.45 \times 10^{12} kilometers, how many light-years away is it?
  5. The quotient (9 \times 10^{8}) \div (3 \times 10^{2}) is typed into a calculator as 9 \times 10^{8} \div 3 \times 10^{2}, without brackets, and the screen shows 3\text{E}10. What did the calculator compute, and what is the quotient?

Reasoning

  1. Explain why the exponents are added when two numbers in scientific notation are multiplied, but must be made equal, and then kept, when two such numbers are added. Use 10^{4} + 10^{4} to show that adding the exponents in a sum gives a wrong answer.
  2. Explain why, after multiplying or dividing two numbers in scientific notation, the leading part never needs more than one shift to get back between 1 and 10, but after subtracting it can need several. Give an example of a subtraction that needs more than one.
  3. Kaveh says that 5 \times 10^{12} + 3 \times 10^{2} is, for any practical purpose, just 5 \times 10^{12}. Work out the exact sum in scientific notation and explain whether he is right.
Answers
  1. 8 \times 10^{8}. Leading parts 2 \times 4 = 8, exponents 3 + 5 = 8.
  2. 9 \times 10^{5}. Leading parts 1.5 \times 6 = 9, exponents -4 + 9 = 5.
  3. 4 \times 10^{5}. Leading parts 5 \times 8 = 40, exponents 7 + (-3) = 4, so 40 \times 10^{4}; one ten moves to the power.
  4. 4.6 \times 10^{-13}. Leading parts 9.2 \times 5 = 46, exponents -6 + (-8) = -14, so 46 \times 10^{-14} = 4.6 \times 10^{-13}.
  5. 2.1 \times 10^{7}. Leading parts 6.3 \div 3 = 2.1, exponents 11 - 4 = 7.
  6. 5 \times 10^{-8}. Leading parts 4.5 \div 9 = 0.5, exponents -2 - 5 = -7, so 0.5 \times 10^{-7}; one ten comes out of the power.
  7. 3 \times 10^{8}. Leading parts 2.4 \div 8 = 0.3, exponents 3 - (-6) = 9, so 0.3 \times 10^{9} = 3 \times 10^{8}.
  8. 4 \times 10^{-4}. The top is 24 \times 10^{3}, and 24 \div 6 = 4 with exponents 3 - 7 = -4.
  9. 4 \times 10^{2} seconds, which is 400 seconds or about 6.7 minutes. \displaystyle \frac{10^{12}}{2.5 \times 10^{9}} = 0.4 \times 10^{3}, writing 10^{12} as 1 \times 10^{12}; then 400 \div 60 \approx 6.7.
  10. 1.08 \times 10^{9} km. (3 \times 10^{5})(3.6 \times 10^{3}) = 10.8 \times 10^{8}, and one ten moves to the power.
  11. 1.2 \times 10^{5} red blood cells. (4.8 \times 10^{6})(2.5 \times 10^{-2}) = 12 \times 10^{4}.
  12. 5 \times 10^{-10}. The missing number is \displaystyle \frac{2 \times 10^{-3}}{4 \times 10^{6}} = 0.5 \times 10^{-9}.
  13. 8.7 \times 10^{9}. Same power: 3.6 + 5.1 = 8.7.
  14. 4.32 \times 10^{6}. 7 \times 10^{4} = 0.07 \times 10^{6}, and 4.25 + 0.07 = 4.32.
  15. 8.5 \times 10^{-3}. 6 \times 10^{-4} = 0.6 \times 10^{-3}, and 9.1 - 0.6 = 8.5.
  16. 1.24 \times 10^{6}. 7.8 + 4.6 = 12.4, so 12.4 \times 10^{5}; one ten moves to the power.
  17. 2.5 \times 10^{7}. 2.05 - 1.8 = 0.25, so 0.25 \times 10^{8}; one ten comes out of the power.
  18. 3 \times 10^{10} cubic meters, which is 30 cubic kilometers. 3.4 \times 10^{9} = 0.34 \times 10^{10}, and 0.34 + 2.66 = 3. Then \displaystyle \frac{3 \times 10^{10}}{10^{9}} = 3 \times 10^{1}.
  19. 3.25 \times 10^{12} bytes, which is 3250 gigabytes. 4 \times 10^{12} - 0.75 \times 10^{12} = 3.25 \times 10^{12}, and dividing by 10^{9} gives 3.25 \times 10^{3}.
  20. 3.7 \times 10^{-6} m, which is 0.000\,003\,7 m, or 3.7 micrometers. Dividing by 10^{-6} leaves the leading part, 3.7.
  21. 20 light-years. \displaystyle \frac{1.89 \times 10^{14}}{9.45 \times 10^{12}} = 0.2 \times 10^{2} = 20.
  22. Working left to right, the calculator computed \left((9 \times 10^{8}) \div 3\right) \times 10^{2} = 3 \times 10^{10}, so 10^{2} multiplied instead of dividing. The quotient is 3 \times 10^{6}: 9 \div 3 = 3 and 8 - 2 = 6.
  23. A product (a \times 10^{m})(b \times 10^{n}) can be reordered and regrouped into (ab) \times 10^{m} \times 10^{n}, and m factors of ten followed by n more is m + n factors, so the exponents add. A sum has no such regrouping: a \times 10^{n} + b \times 10^{n} becomes (a + b) \times 10^{n} only because 10^{n} is a factor of both terms, which needs the powers to be the same, and the shared factor is kept. For instance 10^{4} + 10^{4} = 20\,000 = 2 \times 10^{4}, while 10^{8} = 100\,000\,000.
  24. Each leading part is at least 1 and less than 10. A product of two is at least 1 and less than 100, so it is in range or needs one shift down. A quotient is more than \displaystyle \frac{1}{10} and less than 10, so it is in range or needs one shift up. A difference can be as small as you like, so it may need many shifts: 6.02 \times 10^{8} - 5.99 \times 10^{8} = 0.03 \times 10^{8} = 3 \times 10^{6} needs two.
  25. Exactly, the sum is 5.000\,000\,000\,3 \times 10^{12}, since 3 \times 10^{2} = 0.000\,000\,000\,3 \times 10^{12}. Kaveh is right unless the numbers are known to about eleven digits or more: the change is in the eleventh digit, 300 out of 5 \times 10^{12}, six parts in a hundred thousand million, far smaller than the rounding in any measured quantity.
kg1128 · updated 2026-10-09 · CC-BY-SA-4.0