# Exponents

**Natural number exponents:**

*x*^{n}* = xx...x* (n factors of x)

**example: ***x*^{6}*=x.x.x.x.x.x*

*10x*^{3}*y*^{4}*z*^{2}*=10x.x.x.y.y.y.y.z.z*

**Zero as an exponent:**

*x*^{0}* = 1 *for x any nonzero real number. *0*^{0} is not defined.

**Negative integer exponents:**

*0*^{-n} is not defined for any positive n

**example:**

**Rational number exponents:**

*x*^{1/n} is called the *n*^{th} root of *x* for integer *n > 1* , also written as

and is defined as:

1. For odd integers n > 1, the nth root is a unique real number which when raised to the power of n, gives x

2. For even integers n>1 :

a. If *x > 0*, then the *n*^{th} root is a positive real number which when raised to the power of *n*, gives *x*

b. If *x=0*, the *n*^{th} root of x is *0*

c. If *x < 0*, the *n*^{th} root of *x* for even integer *n* is not a real number (it is a complex number)

*x*^{m/n} is defined as *(x*^{1/n}*)*^{m} if *x*^{1/n} is a real number

**example:**

*4*^{1/2}* = 2*

*(-4)*^{1/2 }*is not a real number*

*8*^{2/3 }*= (8*^{1/3}*)*^{2}*=2*^{2}*=4*

**Laws of exponents and radicals:**

**Scientific notation: **The scientific notation is a way to write more conveniently numbers, that are too long when written in decimal form.

**example: **The number** ***6,890,000,000* in decimal formis written in scientific notation as *6.89 x 10*^{9}

The number *0.000 008 92* is written in scientific notation as *8.92.10*^{-6}

In scientific notation any number is represented as *a x 10*^{b }where *a* is decimal and is called significand or mantissa (which means “meaningful” part), and *b* is an integer exponent. When the absolute value of *a* is between *1* and *10*, as in the above examples, the representation is called **normalized scientific notation**.

The scientific notation represents the “meaningful” part of the number in an easily readable form as a decimal number with absolute value between *1* and *10*, and then multiplies it to the appropriate power of *10* (same as moving the decimal point left or right ) so the original value is not changed. Note in the above example, that the scientific notation is logically similar to reading the number as “six-dot-eighty-nine billion” (where “six-dot-eighty-nine” is the significand or mantissa and “billion” tells us that we have to move the decimal point 9 positions to the right as the original value is “large”). The second example can be read as “eight-dot-ninety-two millionth” and the same considerations apply, only in this case we move the decimal point 6 positions to the left as the original value is “small”. The words “large” and “small” are in quotes here as they have no matematical meaning and we use them only intuitivelly.

**Converting a number in scientific notation** – for example *0.000 000 000 043* :

Determine the significand / mantissa:

Find the “meaningful” part (non-zero digits) and write it placing the decimal point after the first digit: in our example this will give you 4.3 (the significand or mantissa). Preserve the positivity or negativity of the original value – positive decimal number will result in a positive mantissa and negative decimal number will result in negative mantissa. Placing the point after the first digit ensures that the mantissa absolute value is between *1* and *10*.

Determine the exponent:

Count how many positions you have to move the decimal point between the scientific notation (significand) decimal point position and the decimal number representation point position. In this example we have to move the decimal point 11 positions to the left, as the original value is “small”. When moving to the left (representing a ”small” number) the exponent will be negative (because we are dividing by *10*, which is same as multiplying by *10*^{-1} several times), when moving to the right (representing a “large” number) , the exponent will be positive. In this case the exponent is *-11*

Then our original number

*0.000 000 000 043*written in scientific notation is*4.3 x 10*^{-11}*.*

**Converting from scientific notation to decimal number.**

This operation is more straightforward – we only have to move the decimal point of the mantissa the number of positions indicated by the exponent (left for negative exponent / “small” number and right for positive exponent / “large” number).

* *

**example:** 1) write *-23 500 000 000* in scientific notation:

the mantissa / “meaningful” part will be *-2.35 *and we have to move the decimal point 10 times to the right to represent the original value, so the scientific notation is *-2.35.10*^{10}

2) Convert *2.7 × 10*^{-4} to decimal notation.

Moving the decimal point 4 places to the left of 2.7 we get *0.000 27*