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SAT Math : Arithmetic

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Example Questions

Example Question #11 : How To Convert Decimals To Scientific Notation

Raise $6 \times 10 ^{-22}$ to the third power and express the result in scientific notation.

Possible Answers:

$0.18 \times 10 ^{-64}$

$1.8 \times 10 ^{-65}$

$2.16 \times 10 ^{-64}$

$18\times 10 ^{-66}$

$216 \times 10 ^{-66}$

Correct answer:

$2.16 \times 10 ^{-64}$

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:

$(6 \times 10 ^{-22})^{3}$

$= 6 ^{3} \times (10 ^{-22} )^{3}$

$=216 \times 10 ^{-22 \cdot 3}$

$=216 \times 10 ^{-66}$

However, since 216 > 10 , this number is not in scientific notation. Adjust by noting that $216 = 2.16 \times 10^{2}$ , then applying the Product of Powers Rule again:

$216 \times 10 ^{-66}$

$= 2.16 \times 10 ^{2} \times 10 ^{-66}$

$= 2.16 \times 10 ^{2+ (-66)}$

$= 2.16 \times 10 ^{-64}$

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Example Question #18 : How To Convert Decimals To Scientific Notation

Express the result in scientific notation:

$(0.000000004) ^{4}$

Possible Answers:

$2.56 \times 10 ^{-30}$

$2.56 \times 10 ^{-34}$

$256 \times 10 ^{-36}$

$256 \times 10 ^{-32}$

$2.56 \times 10 ^{-38}$

Correct answer:

$2.56 \times 10 ^{-34}$

Explanation:

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

First, express 0.000000004 itself as a number in scientific notation. Move the decimal point to the right until it follows the first nonzero digit - the "4" - as seen below:

Scientific

Since the decimal point was moved 9 places to the right to form the number 4, the number, expressed in scientific notation, is $4 \times 10 ^{-9 }$ .

Consequently,

$(0.000000004) ^{4} = (4 \times 10 ^{-9 }) ^{4}$

This can be rewritten applying the Power of a Product Property, as follows:

$(4 \times 10 ^{-9 }) ^{4} = 4 ^{4} \times (10 ^{-9 }) ^{4}$

Applying the Power of a Power Property, we get

$4 ^{4} \times (10 ^{-9 }) ^{4}$

$= 256 \times 10 ^{- 9 \times 4}$

$= 256 \times 10 ^{-36}$

Since 256 > 10 , this number is not in scientific notation. We can adjust this by noting that

$256 = 2.56 \times 100 = 2.56 \times 10 ^{2}$ ,

substituting, and applying the Product of Powers Property:

$256 \times 10 ^{-36}$

$= 2.56 \times 10 ^{2} \times 10 ^{-36 }$

$= 2.56 \times 10 ^{2+ (-36) }$

$= 2.56 \times 10 ^{-34}$

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Example Question #17 : Other Decimals

Express the result in scientific notation: $20,000, 000 ^{3}$ .

Possible Answers:

$8 \times 10 ^{343}$

$8 \times 10 ^{18}$

$8 \times 10 ^{10}$

$8 \times 10 ^{216}$

$8 \times 10 ^{21}$

Correct answer:

$8 \times 10 ^{21}$

Explanation:

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

First, express 20,000, 000 itself as a number in scientific notation. Add 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 7 places to the left to form the number 2, the number, expressed in scientific notation, is $2 \times 10 ^{7}$ .

Consequently,

$20,000, 000 ^{3} = (2 \times 10 ^{7}) ^{3}$

This can be rewritten applying the Power of a Product Property, as follows:

$(2 \times 10 ^{7}) ^{3}= 2^{3} \times ( 10 ^{7}) ^{3}$

Applying the Power of a Power Property, we get

$2^{3} \times ( 10 ^{7}) ^{3} = 8 \times 10 ^{7 \times 3} = 8 \times 10 ^{21}$

This number is in the correct scientific notation, making this the correct response.

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Example Question #20 : How To Convert Decimals To Scientific Notation

Which of the following represents the quotient

$\frac{6 \times 10 ^{10}}{24 \times 10 ^{29}}$

in scientific notation?

Possible Answers:

$0.25 \times 10 ^{-19 }$

$-2.5 \times 10 ^{20}$

$2.5 \times 10 ^{-18}$

$-0.25 \times 10 ^{19}$

$2.5 \times 10 ^{-20 }$

Correct answer:

$2.5 \times 10 ^{-20 }$

Explanation:

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

Split the fraction as follows:

$\frac{6 \times 10 ^{10}}{24 \times 10 ^{29}} = \frac{6 }{24 } \times \frac{ 10 ^{10}}{ 10 ^{29}}$

Apply the Quotient of Powers Rule:

$\frac{6 }{24 } \times \frac{ 10 ^{10}}{ 10 ^{29}} = 0.25 \times 10 ^{10-29} = 0.25 \times 10 ^{-19}$

However, $0.25 \times 10 ^{-19}$ is not in scientific notation, since |0.25| = 0.25 < 1 .

Adjust by noting that $0.25 = 2.5 \times 0.1 = 2.5 \times 10 ^{-1}$ ; substitute and apply the Product of Powers Rule:

$0.25 \times 10 ^{-19}$

$= 2.5 \times 10 ^{-1} \times 10 ^{-19}$

$= 2.5 \times 10 ^{-1+( -19) }$

$=2.5 \times 10 ^{-20 }$

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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:

$2 \times 10 ^{-12}$

$2 \times 10 ^{-11}$

$2 \times 10 ^{-13}$

$-2 \times 10 ^{14}$

$-2 \times 10 ^{13}$

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 #61 : Decimals

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

Possible Answers:

$1.25 \times 10^{23}$

None of the other choices gives the correct response.

$125 \times 10 ^{21}$

$125 \times 10 ^{343}$

$1.25 \times 10^{345}$

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 #61 : Decimals

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

Possible Answers:

$9.01\times10^5$

$901\times10^3$

$9.02\times10^2$

$9.02\times10^{5}$

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 #402 : Arithmetic

Express the result in scientific notation:

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

Possible Answers:

$0.125 \times 10 ^{-24}$

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

$1.25 \times 10 ^{-28}$

$0.125 \times 10 ^{-27}$

$1.25 \times 10 ^{-25}$

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 #61 : Decimals

Express the following number in scientific notation:

3,880,000,000,000

Possible Answers:

$3.88 \times 10^{12}$

$38.8 \times 10^{11}$

$3.88 \times 10^{-12}$

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

$0.388 \times 10^{13}$

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 #62 : Decimals

Express the result in scientific notation:

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

Possible Answers:

None of these

$6 \times 10 ^{21}$

$6 \times 10 ^{9}$

$6 \times 10 ^{336}$

$6 \times 10 ^{8}$

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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SAT Math : Arithmetic

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #11 : How To Convert Decimals To Scientific Notation

Example Question #18 : How To Convert Decimals To Scientific Notation

Example Question #17 : Other Decimals

Example Question #20 : How To Convert Decimals To Scientific Notation

Example Question #61 : Decimals

Example Question #61 : Decimals

Example Question #61 : Decimals

Example Question #402 : Arithmetic

Example Question #61 : Decimals

Example Question #62 : Decimals

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