# Roots, Powers and Scientific Notation

**Powers **

Powers, also called exponents, cause a number to be multiplied by itself the number of times indicated in the exponent. For example, 5^{2,} ,or 5 squared means 5 • 5, or 25. 5^{3}, or 5 cubed, means 5 • 5 • 5, or 125.

** Roots **

The reverse of a power is a root. If you were asked to find the square root of 25, or √25, the answer would be 5. Other roots are shown by putting the number of the power the root was raised to in the root symbol, such as ∛125 = 5

Often a number will not fit into a root evenly. In those cases, you can factor out any perfect roots and leave the unfactorable numbers inside the root symbol. This is called radical form. You can also use your calculator to find the root in decimal form. If you get an answer on your calculator in radical form and want a decimal, hit the toggle key to change it.

**Negative Exponents**

A number raised to a negative exponent will result in the inverse of that number. For example, 5^{-1} is the same as 1/5. 5^{-2} is equal to 1/25.

**Scientific Notation **

Scientists use powers to more easily express very large or very small numbers. In scientific papers, repeatedly writing large numbers like the distance between planets, which can be millions of miles, can make them difficult to read. Scientists devised a method to use the number ten raised to a power to show the numbers more easily. 10^{2} = 10 • 10 = 100. 10^{3} = 1000. Every time you add one to the exponent of ten, you add another zero to the total. Therefore, multiplying a number times ten to a power is shorthand for writing out that many zeros.

In scientific notation, the decimal is moved to the right of the first number, then that number is multiplied by ten to the power of however many times you moved the decimal. For example:

1200 = 1.2 • 10^{3}

9,300,000 = 9.3 • 10^{6}

8,000,000,000 = 8 • 10^{9}

For very small numbers, the decimal is moved the other way to just to the right of the first number, then the exponent of 10 is a negative exponent of how many times the decimal was moved.
0.01 = 1 • 10^{-2} 0.000043 = 4.3 • 10^{-5}

**Exponents and Roots of Integers**

Since multiplying two negative numbers results in a positive number, negative numbers raised to an even power always result in a positive number. For example, -5^{2} = 25 because -5 • -5 = 25. Odd powers result in a value with the same sign at the original number. For example, -3^{3} means -3 • -3 • -3; -3 • -3 is 9, then 9 • -3 is -27. On your TI-30XS calculator, the calculator will try to add the negative sign after solving for the exponent, so it will interpret -5^{2} as -25. To fix this put negative numbers in parentheses, like (-5)^{2}

Since a number squared will never have a negative value, what happens if we take a square root of a negative number? The square root of -4 is neither 2 nor -2, since both of those numbers squared are positive 4. Instead, your result is an imaginary number, usually represented by the variable 'i'. Since imaginary numbers are not real numbers, you can say that they are undefined in the set of real numbers.