### Exponents and Square Roots

Arithmetic problems on the ASVAB may deal with exponents and square roots. One or two may even involve scientific notation, which is a way of writing large numbers using exponents.

** Exponents** An

*exponent*is a number that tells how many times another number is multiplied by itself. In the expression 4

^{3}, the 3 is an exponent. It means that 4 is multiplied by itself three times or 4 × 4 × 4. So 4

^{3}= 64. The expression 4

^{2}is read "4 to the second power" or "4 squared." The expression 4

^{3}is read "4 to the third power" or "4 cubed." The expression 4

^{4}is read "4 to the fourth power." In each of these cases, the exponent is called a "power" of 4. In the expression 5

^{2}("5 to the second power" or "5 squared"), the exponent is a "power" of 5.

*Examples*

- 3

^{5}= 3 × 3 × 3 × 3 × 3 = 243

- 6

^{2}= 6 × 6 = 36

**Negative Exponents** Exponents can also be negative. To interpret negative exponents, follow this pattern:

*Examples*

** Multiplying Numbers with Exponents** To multiply numbers with exponents, multiply out each number and then perform the operation.

*Examples*

** Square Roots** The

*square*of a number is the number times the number. The square of 2 is 2 × 2 = 4. The

*square root*of a number is the number whose square equals the original number. One square root of 4 (written ) is 2, since 2 × 2 = 4. –2 is also a square root of 4, since –2 × –2 = 4.

*Examples*

### Scientific Notation

For convenience, very large or very small numbers are sometimes written in what is called *scientific notation*. In scientific notation, a number is written as the product of two factors. The first factor is a number greater than or equal to 1 but less than 10. The second factor is a power of 10. (Recall that in an expression such as 10^{3}, which is read "10 cubed" or "10 to the third power," the exponent 3 is a power of 10.)

For example, here is how to write the very large number 51,000,000 in scientific notation. This number can also be written 51,000,000.00. Move the decimal point to the left until you have a number between 1 and 10. If you move the decimal point 7 places to the left, you have the number 5.1000000 or 5.1. Since you moved the decimal point 7 places, you would have to multiply 5.1 by 10^{7} (a power of 10) to recreate the original number. So in scientific notation, the original number can be written as 5.1 × 10^{7}.

For very small numbers, move the decimal point to the right until you have a number between 1 and 10. Then count the decimal places between the new position and the old position. This time, since you moved the decimal point to the right, the power of 10 is a negative. That is, the exponent is a negative number.

*Examples (Large Numbers)*

- 698,000,000,000 can be written 6.98 × 10

^{11}

- 45,000,000 can be written 4.5 × 10

^{7}

*Examples (Small Numbers)*

- 0.00000006 can be written 6 × 10

^{–8}

- 0.0000016 can be written 1.6 × 10

^{–6}

- To change back to the original number from scientific notation, merely move the decimal point the number of places indicated by the "power of 10" number.

*Example (Large Numbers)*

- 2.35 × 10

^{5}becomes 235,000. The decimal point moved 5 places to the right.

- 6 × 10

^{8}becomes 600,000,000. The decimal point moved 8 places to the right.

*Example (Small Numbers)*

2.3 × 10

^{–4}becomes 0.00023. The decimal point moved 4 places to the left.7 × 10

^{–12}becomes 0.000000000007. The decimal point moved 12 places to the left.

** Multiplying in Scientific Notation** To multiply numbers written in scientific notation, multiply the numbers and add the powers of 10.

*Examples*

(2 × 10

^{8}) × (3.1 × 10^{2}) = 6.2 × 1010(3.3 × 10

^{12}) × (4 × 10^{–3}) = 13.2 × 109Here you are adding a 12 and a –3, giving you 9 for the power of 10. But remember the rule that the decimal point must be after a number that is between 1 and 9, so you have to move the decimal place one more place to the left to put it after the 1, and you have to increase the power of 10 accordingly. So the final result of the multiplication is 1.32 × 10

^{10}.

** Dividing in Scientific Notation** To divide in scientific notation, divide the numbers and subtract the powers of 10.

**Examples**

(9 × 10

^{7}) ÷ (3 × 10^{3}) = 3 × 10^{4}Here you divide 9 by 3 (= 3), and you subtract the exponent 3 from the exponent 7 to get the new exponent 4.(1.0 × 10

^{4}) ÷ (4 × 10^{–3}) = 0.25 × 10^{7}Since the number in the first position must be between 1 and 9, move the decimal to the right one place and decrease the exponent accordingly, so the final result is 2.5 × 10

^{6}.