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SAT Math : Other Decimals

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Example Question #61 : Decimals

Which of the following represents the quotient

$\frac{2.5 \times 10 ^{-17}}{12.5 \times 10^{-6}}$

in scientific notation?

Possible Answers:

$2 \times 10 ^{-13}$

$-2 \times 10 ^{13}$

$-2 \times 10 ^{14}$

$2 \times 10 ^{-12}$

$2 \times 10 ^{-11}$

Correct answer:

$2 \times 10 ^{-12}$

Explanation:

A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

Split the quotient as follows:

$\frac{2.5 \times 10 ^{-17}}{12.5 \times 10^{-6}} = \frac{2.5 }{12.5 } \times \frac{10 ^{-17}}{ 10^{-6}}$

Applying the Quotient of Powers Rule:

$\frac{2.5 }{12.5 } \times \frac{10 ^{-17}}{ 10^{-6}} = 0.2 \times 10 ^{-17 - (-6)} = 0.2 \times 10^{-11}$

However, $0.2 \times 10^{-11}$ is not in scientific notation, since -1 < 0.2 < 1 . Adjust by noting that $0.2 = 2 \times 0.1 = 2 \times 10 ^{-1}$ ; substituting and applying the Product of Powers Rule:

$0.2 \times 10^{-11} = 2 \times 10 ^{-1} \times 10^{-11} = 2 \times 10 ^{-1+ (-11)} =2 \times 10 ^{-12}$

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Example Question #62 : Decimals

Which of the following represents the cube of $5 \times 10 ^{7}$ in scientific notation?

Possible Answers:

$125 \times 10 ^{21}$

$1.25 \times 10^{23}$

None of the other choices gives the correct response.

$1.25 \times 10^{345}$

$125 \times 10 ^{343}$

Correct answer:

$1.25 \times 10^{23}$

Explanation:

A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

To find $(5 \times 10 ^{7})^{3}$ , apply the Power of a Product Rule, then the Product of Powers Rule, as follows:

$(5 \times 10 ^{7})^{3}$

$= 5 ^{3} \times (10 ^{7} )^{3}$

$=125 \times 10 ^{7 \cdot 3}$

$=125 \times 10 ^{21}$

However, since 125> 10 , this number is not in scientific notation. Adjust by noting that $125 = 1.25 \times 100 = 1.25 \times 10^{2}$ , then applying the Product of Powers Rule again:

$125 \times 10 ^{21}$

$=1.25 \times 10^{2} \times 10 ^{21}$

$=1.25 \times 10^{2+21}$

$=1.25 \times 10^{23}$ ,

the correct response.

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Example Question #63 : Decimals

Round 901,527 to the nearest thousand and convert to scientific notation.

Possible Answers:

$9.02\times10^2$

$9.01\times10^5$

$901\times10^3$

$9.02\times10^{5}$

Correct answer:

$9.02\times10^{5}$

Explanation:

When rounding to the nearest thousand, look to the hundreds place to determine whether you need to round up or down. You always round down when the digit is between 0 and 4, and up when it is between 5-9. Therefore, the rounded number is 902,000. When using scientific notation, the first number in the notation must be less than 10. In this case, that number is 9.

From there, the decimal goes immediately after. Then count how many places the decimal would have to be moved in order to convert back to the original number (5 places).

$9.02\times10^{5}$

When the decimal is moved to the left when writing it in scientific notation, the exponent is positive. When moved to the right to write the number in scientific notation, the exponent is negative.

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Example Question #64 : Decimals

Express the result in scientific notation:

$\left (\frac{1}{200,000,000 } \right )^{3}$

Possible Answers:

$1.25 \times 10 ^{-25}$

$1.25 \times 10 ^{-28}$

$0.125 \times 10 ^{-24}$

$0.125 \times 10 ^{-27}$

$\textup{None of the other choices gives the correct response}$

Correct answer:

$1.25 \times 10 ^{-25}$

Explanation:

First, rewrite 200,000,000 as a number in scientific notation. A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

200,000,000 can be rewritten by adding the implied decimal point to the end of the number, and move it to the left until it follows the first nonzero digit - the "2" - as seen below:

Scientific

Since the decimal point was moved 8 places to the left to form the number 2, the number, expressed in scientific notation, is $2 \times 10 ^{8 }$ .

Therefore, $\frac{1}{200,000,000 } = \frac{1}{2 \times 10 ^{8 }}$ . By the Negative Exponent Rule ,

$\frac{1}{a} = a^{-1}$ , so

$\frac{1}{2 \times 10 ^{8 }} =( 2 \times 10 ^{8 } )^{-1}$

Applying the Power of a Product Property, we get

$( 2 \times 10 ^{8 } )^{-1} = 2^{-1} \times ( 10 ^{8 } )^{-1}$

Applying the Negative Exponent Rule and the Power of a Product Property on the right, we get

$2^{-1} \times ( 10 ^{8 } )^{-1} = \frac{1}{2} \times 10 ^{8 \cdot (-1)} = 0.5 \times 10 ^{-8 }$

Therefore, $\frac{1}{200,000,000 }= 0.5 \times 10 ^{-8 }$ .

$\left (\frac{1}{200,000,000 } \right )^{3 }=( 0.5 \times 10 ^{-8 })^{3}$

Applying the Power of a Product Property:

$( 0.5 \times 10 ^{-8 })^{3} = 0.5^{3} \times (10 ^{-8 })^{3}$

Applying the Product of Powers Property:

$0.5^{3} \times (10 ^{-8 })^{3} = 0.125 \times 10 ^{-8 \times 3} = 0.125 \times 10 ^{-24}$

Since 0< 0.125 < 1 , this is not in scientific notation; adjust it by noting that

$0.125 = 1.25 \times 0.1 = 1.25 \times 10 ^{-1}$

substituting, and applying the Product of Powers Property:

$1.25 \times 10 ^{-1} \times 10 ^{-24} = 1.25 \times 10 ^{-1+ (-24)} = 1.25 \times 10 ^{-25}$

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Example Question #65 : Decimals

Express the following number in scientific notation:

3,880,000,000,000

Possible Answers:

$3.88 \times 10^{12}$

$3.88 \times 10^{-12}$

$38.8 \times 10^{11}$

$0.3 88 \times 10^{-13}$

$0.388 \times 10^{13}$

Correct answer:

$3.88 \times 10^{12}$

Explanation:

A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

To convert 3,880,000,000,000 to scientific notation, place the implied decimal point after the final zero and move it to the left as many places as is necessary until it is after the first nonzero digit - in this case the "3". Note that the point is moved 12 places to the left.

Scientific

The number in front is 3.88, the number formed. The exponent of 10 is 12 - positive since the point was moved to the left. Therefore, the number, in scientific notation, is $3.88 \times 10^{12}$ .

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Example Question #66 : Decimals

Express the result in scientific notation:

$1 \times 10 ^{8} + 2 \times 10 ^{7} + 3 \times 10 ^{6}$

Possible Answers:

None of these

$6 \times 10 ^{8}$

$6 \times 10 ^{21}$

$6 \times 10 ^{336}$

$6 \times 10 ^{9}$

Correct answer:

None of these

Explanation:

A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

An easy way to add these numbers is to note that if and are both positive integers, $a \times 10^{N}$ is the number followed by zeroes. Therefore,

$1 \times 10 ^{8} = 100,000,000$

$2 \times 10 ^{7} = 20,000,000$

$3 \times 10 ^{6} = 3,000,000$

Add these numbers:

$\begin{matrix} 100,000,000\\ \; \; 20,000,000\\ \; \; \; \underline{3,000,000}\\ 123,000,000 \end{matrix}$

Since all four choices can be rewritten as 6 followed by a number of zeroes, none of them are equal to this sum.

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Example Question #21 : How To Convert Decimals To Scientific Notation

Which of the following represents the product of

$(4 \times 10 ^{22}) (7 \times 10 ^{12})$

in scientific notation?

Possible Answers:

$28 \times 10^{34}$

None of these

$28 \times 10^{264}$

$2.8 \times 10^{35}$

$2.8 \times 10^{265}$

Correct answer:

$2.8 \times 10^{35}$

Explanation:

A number in scientific notation takes the form $a \times 10^{N}$ , where $1 \le |a| < 10$ and is an integer.

To multiply two numbers that are in scientific notation, first, use commutativity to multiply the numbers:

$(4 \times 10 ^{22}) (7 \times 10 ^{12})$

$= 4 \times 7 \times 10 ^{22} \times 10 ^{12}$

Applying the Product of Powers Rule on the powers of 10:

$4 \times 7 \times 10 ^{22} \times 10 ^{12}$

$= 28 \times 10 ^{22+12}$

$= 28 \times 10 ^{34}$

However, since 28 > 10 , this number is not in scientific notation. Adjust by noting that $28 = 2.8 \times 10 = 2.8 \times 10^{1}$ , then applying the Product of Powers Rule again:

$28 \times 10 ^{34}$

$= 2.8 \times 10^{1} \times 10 ^{34}$

$= 2.8 \times 10^{1+ 34}$

$= 2.8 \times 10^{35}$

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Example Question #1 : How To Convert Decimals To Scientific Notation

Convert 0.0004640 into scientific notation.

Possible Answers:

$4.64 \times 10^4$

$4.64\times10^{-4}$

The value is already in scientific notation

4640

$4640\times10^{-4}$

Correct answer:

$4.64\times10^{-4}$

Explanation:

When written in scientific notation, a number will follow the format x*10^y in which is between one and ten and is an integer value.

To find , take the first non-zero digit in your given number as the ones place. In 0.0004640 this would be the first 4. All subsequent digits fall into the tenths, hundredths, etc. places.

x=4.640

To find , we must count the number of places that is removed. In 0.0004640, the first digit of is in the ten-thousandths place. This indicates that will be .

Together, the final scientific notation will be $4.640*10^{-4}$ .

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Example Question #1 : How To Add Decimals

If Johnny buys two comic books, priced at $1.50 each, and a candy bar, priced at $0.75, he'll have three quarters and two dimes left over. How much money does he have right now?

Possible Answers:

$5.10

$4.70

$4.35

$3.75

$3.20

Correct answer:

$4.70

Explanation:

Add what he can purchase to what he has left over:

Two comic books and the candy bar: $1.50 + $1.0 + $0.75 = $3.75

Three quarters and two dimes: $0.75 + $0.20 = $0.95

Therefore his total amount of money is $3.75 + $0.95 = $4.70.

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Example Question #1 : How To Add Decimals

Add: 0.4+0.005

Possible Answers:

0.45

0.009

0.4005

0.405

0.9

Correct answer:

0.405

Explanation:

In order to add the decimals, add placeholders to the decimal 0.4 .

Be careful not to add the wrong digits!

0.4+0.005 = 0.400+0.005

Add the thousandths places.

0+5 = 5

Add the hundredths places.

0+0=0

Add the tenths places.

4+0=4

Combine the numbers and put a decimal before the tenths place.

The correct answer is: 0.405

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SAT Math : Other Decimals

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #61 : Decimals

Example Question #62 : Decimals

Example Question #63 : Decimals

Example Question #64 : Decimals

Example Question #65 : Decimals

Example Question #66 : Decimals

Example Question #21 : How To Convert Decimals To Scientific Notation

Example Question #1 : How To Convert Decimals To Scientific Notation

Example Question #1 : How To Add Decimals

Example Question #1 : How To Add Decimals

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