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Math 125 Exponents

Exponents are numbers that indicate a repeated factor. It tells you to a multiply a
number times itself a certain number of times. For example:

43 = x 43 is read as 4 to the third power. Usually a
number to the third power is said to be cubed.
So you would read this as four cubed. Four is
the factor and three is the exponent. The exponent
tells you how many times to multiply the factor by
4 *4 *4 is the same as 43  Now simplify. Remember to do
multiplication from left to right.
16 * 4
64 = x
43 is 64.

How to read exponents:

42 is read as four squared.
43 is read as four cubed.
44 is read as four to the fourth.
45 is read as four to the fifth.
. . .


~ A number (x) to the power zero (0) is one (1). 50 = 1
~ A number (x) to the power one (1) is the number (x). 51 = 5

When adding/subtracting numbers with exponents, just
add/subtract the coefficients. Note: The powers on the
variables MUST be the SAME.
Solution: 3x2
The exponents are not the same. One is cubed and the other
is to the fifth power. So they cannot be added together.
  Solution: x3 + 5x5
When multiplying numbers with exponents, just multiply
the coefficients. Then, carry over the variable and add
the exponents.
When you multiply the coefficients, you get 2. Then you
carry over the m. Next, add the exponents of the variable
2 + 5 is 7
  Solution: 2m7
When dividing monomials (single terms), just divide the
In division, subtract the powers of like variables. If the
larger exponent is on the top, subtract the bottom from the
top. If the larger exponent is on the bottom, subtract the
top from the bottom. Place the final answer where the
larger exponent was (numerator or denominator).
The variable (r) goes in the numerator because the larger
exponent (5) was in the numerator. Simplify.
5 - 2 is 3.
  Solution: 4r3
Divide coefficients.
Subtract exponents of like variable. n will be in the
denominator because 5 is larger than 4. y will be in
the numerator because 6 is larger than 2.
6 - 2 is 4. 5 - 4 is one. Coefficients one are not written.
Now multiply times 5.
Power to Power:  
Here, a number with a power is being raised to another
power. Raise all coefficients to the outside power. Raise
any variable to the outside power by carrying over the
variable and multiplying the exponents.
2 cubed is 8. 4 times 3 is 12.
  Solution: 8r12

Negative Exponents:

~ If a term in the numerator has a negative exponent, move it to the
denominator. Change the negative sign to a positive.

~ If a term in the denominator has a negative exponent, move it to the
numerator. Change the negative sign to a positive.

Note: If after all negative exponents are moved there are no terms left in either the
numerator or the denominator, then put a 1 there.

Move the y-5 to the denominator to make the exponent
The exponent became positive and there were no terms
in the numerator, so one is placed there. Now multiply
like terms.

Scientific Notation: Numbers in scientific notation are expressed as


~ In scientific notation a has to be a whole number greater than or equal to1 and
less than 10, and b can be any number.

~ If the decimal point is not shown, it is to the right side of the ones digit.

~ If the original number is greater that 10, the exponent (x) is positive. If it is
less than 1 it is negative.

Standard Notation to Scientific Notation

1) Express 800 in scientific notation. To express 800 in
scientific notation, you have to get a.
To get a, you move the decimal
point, so that the number becomes
an integer between 0 and 10. You
have to move the decimal point 2
places to the left to make the
number an integer between 0 and
10. The amount of places you move
the decimal point is the exponent of
the 10, so the scientific notation for
800 is


2) Express .0098 in scientific notation.


To get a, you have to move the
decimal point three places to the
right, so the number becomes
9.8 (an integer between 0 and
10). Since the decimal was moved
three places to the right, the
exponent of the ten is –3, so the
scientific notation for .0098 is


Scientific Notation to Standard Notation


Using this formula, determine what a
is. In this problem a is 9. x tells how many
spaces to move the decimal point of a. If x is
positive, move the decimal point to the right.
If x is negative, move the decimal point to the
left. x is –4. So, move the decimal point of a
four spaces to the left.

Solution: .0009