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Scientific Notation: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 A place value record

A whole number has $2$ in its millions place, $9$ in its ten-thousands place, and zeros in all its other places. Write the number in scientific notation.
1. Hint 1
  The position of the first nonzero digit determines the power of ten.
2. Hint 2
  Keep the zero between the $2$ and the $9$ ; the zeros after the $9$ are dropped.
Answer

$2.09\times10^6$ .

Full solution

The two place values give the ordinary number $2{,}000{,}000+90{,}000$ , which is $2{,}090{,}000$ .

The coefficient must have one nonzero digit before its decimal point, so it is $2.09$ .
The zero between the $2$ and the $9$ stays, and the zeros after the $9$ are dropped.

The point moves six places left, from after the last zero to just after the $2$ , so the exponent is $6$ .
$2{,}090{,}000=2.09\times10^6$

Answer

$2.09\times10^6$ .

Key idea

A zero between nonzero digits stays in the coefficient; zeros after the last nonzero digit are dropped.
Problem 1: Mark complete
Problem 2 Where a digit sits

When $6.03\times10^{-7}$ is written as an ordinary decimal, in which decimal place does the digit $3$ sit?
1. Hint 1
  The exponent fixes the decimal place occupied by the leading digit.
2. Hint 2
  Write the number out in full by moving the decimal point of $6.03$ the number of places the exponent gives, then count decimal places up to the $3$ .
Answer

The ninth decimal place.

Full solution

The exponent $-7$ moves the decimal point of $6.03$ seven places left.
$6.03\times10^{-7}=0.000000603$

The leading $6$ sits in the seventh decimal place.
The $0$ follows it in the eighth place, and the $3$ sits in the ninth place.

Answer

The ninth decimal place.

Key idea

In scientific notation, an exponent of $-n$ puts the leading digit in the $n$ th decimal place, and the coefficient's other digits, zeros included, follow it in order.
Problem 2: Mark complete
Problem 3 A missing coefficient

Fill the box in $0.000024=\square\times10^{-5}$ .
1. Hint 1
  The power of ten tells you how much the coefficient is being multiplied by.
2. Hint 2
  Find the number that multiplies $0.00001$ to give $0.000024$ .
Answer

$2.4$ .

Full solution

The power of ten is
$10^{-5}=0.00001.$
Divide the original value by that factor.
$0.000024\div0.00001=2.4$

The coefficient is in the required scientific-notation range.
Checking the product gives
$2.4\times10^{-5}=0.000024$

Answer

$2.4$ .

Key idea

The coefficient can be recovered by dividing a number by its power-of-ten factor.
Problem 3: Mark complete
Problem 4 A file transfer

A file contains $3.2\times10^4$ bytes, including a header of $8\times10^2$ bytes. The data after the header is sent in packets of $2\times10^2$ bytes each. How many packets are needed?
1. Hint 1
  Only the bytes after the header are divided into packets.
2. Hint 2
  Express the file and header sizes as ordinary numbers before subtracting.
3. Hint 3
  Divide the remaining data size by the size of one packet.
Answer

$156$ packets.

Full solution

The file contains $32{,}000$ bytes and the header contains $800$ bytes.
Subtract to find the data size.
$32{,}000-800=31{,}200$

Each packet holds $200$ bytes.
$31{,}200\div200=156$

Exactly $156$ full packets are needed.
Checking, $156\times200=31{,}200$ bytes, and adding the header returns the file size.

Answer

$156$ packets.

Key idea

Convert representations consistently before subtracting a part and dividing the remainder into equal groups.
Problem 4: Mark complete
Problem 5 A printed design

A printer places $2.4\times10^3$ dots along each cm of a line. A design contains three lines, each $3.5\times10^{-2}$ cm long. How many dots are printed in total? Give the count in scientific notation and as an ordinary whole number.
1. Hint 1
  Find the dot count for one line, then account for all three lines.
2. Hint 2
  Multiply the number of dots per cm by the length, combining their coefficients and powers of ten.
3. Hint 3
  After multiplying by three, check that the coefficient is at least $1$ and less than $10$ , and renormalize if it is not.
Answer

$2.52\times10^2$ dots, and $252$ dots as an ordinary whole number.

Full solution

For one line, the coefficients multiply to $2.4\times3.5=8.4$ , and the exponents add.
$10^3\times10^{-2}=10^1$

For three lines, multiply the coefficient by three.
$3\times8.4=25.2$

The total is $25.2\times10^1$ dots.
Put the coefficient in range by moving a factor of ten into the power.
$25.2\times10^1=2.52\times10^2$

As an ordinary count,
$2.52\times10^2=252$

The design contains $252$ dots.

Answer

$2.52\times10^2$ dots, and $252$ dots as an ordinary whole number.

Key idea

To multiply in scientific notation, multiply the coefficients, add the exponents, and renormalize the coefficient at the end.
Problem 5: Mark complete
Problem 6 Batches into portions

Each of seven batches contains $3\times10^{-1}$ liter of liquid. Together, they fill $3\times10^4$ equal portions. Find the volume of one portion in scientific notation.
1. Hint 1
  Find the total volume from all seven batches before dividing it into portions.
2. Hint 2
  Divide the coefficients, and subtract the exponent of the portion count from the exponent of the total.
3. Hint 3
  Make sure the final coefficient is at least $1$ and less than $10$ .
Answer

$7\times10^{-5}$ liter.

Full solution

The seven batches hold $7\times(3\times10^{-1})$ liters.
Multiplying the numbers in front of the power of ten gives
$7\times3=21$

The total volume is $21\times10^{-1}$ liters.
Move one factor of ten into the power to bring the coefficient into range.
$21\times10^{-1}=2.1\times10^0$

That is $2.1$ liters.
Divide it by $3\times10^4$ portions by dividing the coefficients and subtracting the exponents.

The coefficients give
$2.1\div3=0.7$

The exponents give
$0-4=-4$

This gives $0.7\times10^{-4}$ liter.
The coefficient is below $1$ , so multiply it by ten and decrease the exponent by one.
$0.7\times10^{-4}=7\times10^{-5}$

One portion contains $7\times10^{-5}$ liter.
Checking, $3\times10^4$ portions of that size hold $21\times10^{-1}$ liters, or $2.1$ liters, returning the total volume.

Answer

$7\times10^{-5}$ liter.

Key idea

Find a combined quantity before dividing its coefficient and power of ten among equal portions.
Problem 6: Mark complete
Problem 7 Stored record sizes

Four records have sizes $32{,}000$ , $3.1\times10^4$ , $2.9\times10^5$ , and $9\times10^3$ bytes. List the sizes from smallest to largest, writing every size in scientific notation.
1. Hint 1
  Rewrite $32{,}000$ with a coefficient at least $1$ and less than $10$ , so all four sizes share one form.
2. Hint 2
  Compare the exponents first; compare coefficients where exponents tie.
Answer

$9\times10^3$ , $3.1\times10^4$ , $3.2\times10^4$ , $2.9\times10^5$ bytes, in that order.

Full solution

The ordinary whole-number size converts to
$32{,}000=3.2\times10^4$

The exponent $3$ gives the smallest size and the exponent $5$ gives the largest.
The two middle sizes have exponent $4$ , so their coefficients decide the order.
$3.1<3.2$

The ascending order is $9\times10^3$ , $3.1\times10^4$ , $3.2\times10^4$ , then $2.9\times10^5$ bytes.

Answer

$9\times10^3$ , $3.1\times10^4$ , $3.2\times10^4$ , $2.9\times10^5$ bytes, in that order.

Key idea

Once every coefficient is at least $1$ and less than $10$ , compare powers of ten first and use coefficients only to settle equal exponents.
Problem 7: Mark complete
Problem 8 A marked interval

A number line has marks at $-0.000001$ , zero, and $0.000001$ . Sam places $9\times10^{-8}$ between zero and the positive mark. Is the placement correct? Explain.
1. Hint 1
  A negative exponent on ten does not turn a positive coefficient into a negative number.
2. Hint 2
  Write the value and the positive mark with the same power of ten, or as ordinary decimals.
Answer

Yes. $0<0.00000009<0.000001$ .

Full solution

The indicated value is a small positive number.
$9\times10^{-8}=0.00000009$

The positive mark is $1\times10^{-6}$ .
Its exponent is larger than $-8$ , and both coefficients are in range, so it is larger than $9\times10^{-8}$ .
The value lies between zero and the positive mark, so Sam's placement is correct.

Answer

Yes. $0<0.00000009<0.000001$ .

Key idea

A negative power-of-ten exponent can place a positive number very close to zero.
Problem 8: Mark complete
Problem 9 A recorder rule

A recorder accepts entries $a\times10^n$ with integer $n$ and $1\le a\le10$ , including both endpoints. Decide whether some positive number has two different accepted entries, and support your decision with an example or a reason. Then explain how requiring $a<10$ changes the situation.
1. Hint 1
  The two endpoints of the allowed coefficient range differ by a factor of ten.
2. Hint 2
  A factor of ten can be transferred between the coefficient and the power.
3. Hint 3
  Test whether an entry whose coefficient is exactly $10$ has another accepted form.
Answer

Yes, for example $10\times10^{-3}$ and $1\times10^{-2}$ . Requiring $1\le a<10$ gives one form for each positive number.

Full solution

The accepted coefficient $10$ can be moved into the power of ten.
$10\times10^{-3}=1\times10^{-2}$

Both entries equal $0.01$ , so including both endpoints permits two different entries for the same value.
Requiring the coefficient to be less than $10$ excludes the first entry.

In the range $1\le a<10$ , multiplying a coefficient by ten makes it too large, and dividing it by ten makes it too small.
Changing the exponent by more than one multiplies or divides the coefficient by $100$ or more, which moves it even further out of range.
Thus changing the integer exponent cannot produce another coefficient in the allowed range for the same value.

Answer

Yes, for example $10\times10^{-3}$ and $1\times10^{-2}$ . Requiring $1\le a<10$ gives one form for each positive number.

Key idea

Excluding ten from the coefficient range prevents duplicate scientific-notation forms at the boundary.
Problem 9: Mark complete
Problem 10 A revised result

A calculation produces $0.84\times10^5$ . Dev rewrites it as $8.4\times10^6$ . Do the two expressions have the same value? Either way, give the proper scientific-notation form of the original result.
1. Hint 1
  Changing the coefficient must be balanced by an opposite change in the power-of-ten factor.
2. Hint 2
  Write both $0.84\times10^5$ and $8.4\times10^6$ as ordinary numbers and compare them.
Answer

No. The correct form is $8.4\times10^4$ .

Full solution

The original result is
$0.84\times10^5=84{,}000$

Multiplying the coefficient by ten requires reducing the exponent by one.
$0.84\times10^5=8.4\times10^4$

Dev increased both parts instead.
His expression is
$8.4\times10^6=8{,}400{,}000$

The values are different.
The corrected coefficient $8.4$ is in the required range.

Answer

No. The correct form is $8.4\times10^4$ .

Key idea

Increasing the coefficient by a factor of ten requires decreasing the exponent by one to preserve the value.
Problem 10: Mark complete

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