PSAT Math - How to convert decimals to scientific notation

can be represented by which of the following notations?

$4.632 \times 10^{-3}$

$46.32 \times 10^-3$

$4.632 \times 10^{-4}$

$4.632 \times 10^3$

$463.2 \times 10^2$

Explanation

In order to convert the decimal to scientific notation, count the number of spaces from the decimal point to the space between the 4 and the 6 (scientific notation requires that the leftmost number be a number between 1 and 10). In this instance it is 3 spaces. Since the decimal point will be moving from left to right, we know that our exponent will be negative, and in this case -3.

then becomes X $10^{-3}$ .

Define an operation $\bigstar$ as follows:

For all real numbers ,

$a\bigstar b= a + b^{2}$

Which of the following is equal to $0.0004\bigstar 0.003$ ?

$4.09 \times 10^{-4}$

$4.9 \times 10 ^{-7}$

$1.156 \times 10^{-5}$

$1.156 \times 10^{-4}$

$4.9 \times 10^{-4}$

Explanation

Substitute and execute the operations:

$a\bigstar b= a + b^{2}$

$0.0004\bigstar 0.003 = 0.0004 + 0.003^{2}$

Move the decimal point right four places to get it after the first nonzero digit; the scientific notation equivalent is

$4.09 \times 10^{-4}$

Express 770,000,000,000 in scientific notation.

$7.7 \times 10^{11}$

$77 \times 10^{11}$

$77 \times 10^{10}$

$7.7 \times 10^{10}$

$0.77 \times 10^{12}$

Explanation

Write the number with a decimal point and without commas:

Counting the number of places, move the decimal point to the left until it is after the first nonzero digit:

The resulting number is 7.7; the number of places the decimal point moved to the left was 11. In scientific notation, this number is $7.7 \times 10^{11}$ .

Convert into scientific notation.

Explanation

To convert a number into scientific notation, the decimal must be moved until our base number is greater than 1 and less than 10. Next, we must count the number of places we moved it and multiply it by 10 to that number. Thus, our answer is:

Define an operation $\bigstar$ as follows:

For all real numbers ,

$a\bigstar b=a - b + 0.00000001$

Which of the following is equal to $0.00005 \bigstar 0.0000006$ ?

$4.941 \times 10 ^{-5}$

$4.939 \times 10 ^{-5}$

$4.401 \times 10 ^{-5}$

$4.941 \times 10 ^{-4}$

$4.401 \times 10 ^{-4}$

Explanation

Substitute and execute the operations:

$a\bigstar b=a - b + 0.00000001$

$0.00005 \bigstar 0.0000006 = 0.00005 - 0.0000006 + 0.00000001$

Move the decimal point right five places to get it after the first nonzero digit; the scientific notation equivalent is

$4.941 \times 10 ^{-5}$

Define an operation $\bigstar$ as follows:

For all real numbers ,

$a\bigstar b= a + 0.2b$

Which of the following is equal to $0.004 \bigstar 0.03$ ?

$1 \times 10^{-2}$

$4.6 \times 10 ^{-3}$

$6.4 \times 10^{-2}$

$6.12 \times 10 ^{-2}$

$6.12 \times 10 ^{-3}$

Explanation

Substitute and execute the operations:

$a\bigstar b= a + 0.2b$

$0.004 \bigstar 0.03 = 0.004 + 0.2 \cdot 0.03$

Move the decimal point right two places to get it after the first nonzero digit; the scientific notation equivalent is

$1 \times 10^{-2}$

12 grams of oxygen are equal to $6.022 \times 10^{23}$ atoms. Which of the following comes closest to the mass, in grams, of one million atoms of oxygen?

$1.99 \times 10 ^{-17}$

$1.99 \times 10 ^{-29}$

$7.22 \times 10 ^{-16}$

$7.22 \times 10 ^{-28}$

$5.03 \times 10 ^{-18}$

Explanation

In scientific notation, one million atoms is equal to $1 \times 10^{6}$ atoms. To find the mass of this many atoms, we can solve the proportion statement

$\frac{m}{1 \times 10^{6}} = \frac{12} {6.022 \times 10^{23}}$

$m= \frac{12} {6.022 \cdot 10^{23}} \cdot \left (1 \times 10^{6} \right )$

$m= \frac{12 \cdot 10^{6}} {6.022 \cdot 10^{23}}$

$m= \frac{12} {6.022} \cdot \frac{10^{6}}{10^{23}} \approx 1.99 \times 10 ^{-17}$

Represent $\small 0.000003452$ in scientific notation.

$\small 3.452 \times 10^{-6}$

$\small \small 3.452 \times 10^{-5}$

$\small 3.452 \times 10^{6}$

$\small \small \small 3.452 \times 10^{5}$

$\small \small \small \small 3.452 \times 10^{7}$

Explanation

$\small \small \small \small 0.000003452 = 3.452 \times 10^{-6}$

Trick: Count how many times you move the decimal to get 3.452. We moved the decimal 6 times to the right so the exponent above 10 will be -6. If we move the decimal to the left then our exponent would be positive.

Another way to remember whether the exponent will be negative or possitive is whether or not the original number is less than or greater than one. If the original number is less than one, when you put it in scientific notation the exponent above the 10 will be negative. On the other hand, if the original number is greater than one, then the exponent above the 10 will be positive.

Convert 0.0004640 into scientific notation.

$4640\times10^{-4}$

$4.64\times10^{-4}$

$4.64 \times 10^4$

The value is already in scientific notation

Explanation

When written in scientific notation, a number will follow the format 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.

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 .