Exponents & Roots Flashcards
What must be true for addition and subtraction of exponents?
They must have the same base and exponent.
I.e. 2r^3 + 6r^3 is okay
You cannot do
2r^5 + 2r^7
Or
4z^7 - 4d^2
What must be true in order to multiply exponents?
The numbers must have the same base or exponent
How do you multiply exponents?
if they have the same base, you add their exponents and multiply their coefficients
I.e. 4x^2 * 9x^5=36x^7
If they do not have the same base do not add their exponents but do multiply their coefficients
If the term has the same base and exponents you can add their exponents OR multiply their bases
How do you multiply an exponent with the same base and same exponent?
Give an example of a problem
You can either add their exponents OR multiply their bases
I.e. 5^x * 5^x = 25x or 5^2x
How do you multiply exponential terms with the same base ?
Give an example.
You add their exponents and multiply their coefficients
I.e. 3z^2 x 4z^3= 12z^5
How do you multiply an exponential term with the same exponent?
Give an example.
You multiply their bases and multiply their coefficients
I.e. 7d^4 * 5d^4= 35d^4
How do you divide a term with an exponent if it has the same base?
What do you do if it has a coefficient?
Give An example
You subtract their exponents
Do not divide their bases but do divide their coefficients
z^7 / z^5 = z^2
How do you divide a term that has the same exponent?
What if it has a coefficient?
Give an example.
Divide their bases and divide their coefficients
I.e. 9^4 / 3^4= 3^4
What is the answer to a division equation where there is a term with an exponent with the same base and exponent?
The quotient if their coefficients times 1
I.e. 5^3 / 5^3= 1
8x^4 / 4x^4=2
What do you do with negative exponents?
What if they have a coefficient?
Flip the base and make the exponent positive
Leave the coefficients as they are
Does the exponent apply to the base and coefficient of a term?
No, only the base
How do you apply fractional roots?
Take the base to the root of the denominator
Then
Use the numerator as the power of the new base
How do you solve problems w exponential expressions on both sides of an equations
Make sure that both expressions have the same base or same exponent
Eliminate what is the same
Solve for what is left behind
What can be a key to solving some exponent problems that involve addition and subtraction?
Rewrite the problem by reversing the rule
How do you handle numbers w large exponents or large bases?
You split the exponent or base
Split exponents by having two exponents that add to the current exponent
Split bases by having bases that multiply to the current base
(Most often the bases will need to be broken down)
What should you do if given an exponential expression with a fractional base?
Rewrite the base
An even number times any number results in a ___solution?
Even
All powers of 6 are….?
Even
What is the opposite of a root?
An exponent
A negative root is denoted by what?
A negative radical
What is the square root rule for square 2-10
From sq root of 4 it increases by .2
From sq root of 4 it decreases by .3
How do you simplify a sq root ?
Rewrite the root as a product of its factors inside the radical and simplify any pairs that lie within
When cant a sq root be simplified?
When it does not contain a pair of factors
What should you remember when a root has a coefficient and you are factoring the root?
Multiply the coefficient w the pair you put outside the radical
How do you simplify a simple square root (roots w/o addition or subtraction)
Break up the root
How do you simplify complex roots (roots w addition or subtraction)
combine terms within the radical
What roots can be added or subtracted?
Those w the same radical
How do you multiply square roots?
Combine terms under a single radical (simplify terms to make things easier)
Multiply coefficients together if you have them
How do you divide square roots?
Break down radicals and cancel out like terms
Be sure to keep coefficients separate
Whats the best way to approximate a square root?
Think of squares that the number lies between
How do you get a root out of the denominator in a non complex denominator?
Multiply the top and bottom of the fraction by that root
How do you simplify a complex radical denominator?
Multiply the equation by the conjugate of the denominator
What is key to solving complex square root problems?
Factoring out terms in common within the denominator and looking for common perfect squares once u simplify the terms
Decimal places of answer x root number = decimal places of original
How to calculate roots And decimals
Whats the easiest way to multiply square roots?
Combine the terms under a single radical
What numbers have square roots, cube roots, and quarter roots that are less than 1
Only numbers less than 1
What is 2^29 - 2^28 =
2^28(2-1)= 2^28
How do you solve 101^2 - 99^2 ?
Do difference of squares
(101+99)(101-99)=200(2)=400
If a root has a coefficient it is better to convert the coefficient to a root. Provide an example.
If you have 3 sq rt 7
Turn the 3 into a sq rt of 9
Multiply sq rt 9 by sq rt 7 and gget sq rt 63
Sq rt 63 is a bit less than sq rt 64 which is 8
So sq rt 63 is approx. 7.9
