Powers & Roots
Practice: Just as (ab)^2 = (a^2)(b^2), you need to be able to discern that (a^2)(b^2)=(ab)^2. So when a problem reads (2^-n/3)(3^-n/2), it really can read ...
(2^-n) (3^-n)/ (3)(2) = 6^-n/6
You should know the approximate square root of 0, 1, 2, 3 and 5 which are...
0, 1, 1.4 , 1.7, 2.2
When factoring, remember that every number on the number line has this value as a factor ...
1
1^n and 0^n equals
1 and 0
Know the basic powers of single digits, such as..
1-9 to the power of 1-9
When trying to solve any problem on the GRE, here are the steps you to should follow ....
1. Analyze the question 2. Identify the task, Approach strategically 3. Confirm your answer.
When dealing with the "Units" digits problem, the strategy for solving this is to ...
1. Look for a repeating pattern (also referred to as the "period"). 2. Figure out where that pattern will be at the desired power.
B^(-n) or (q/r)^(-n) or 1/(B^-n) is
1/B^n or (r/q)^n or B^n
Remember that in x^2=4, the value of "x" can be either...
2 or -2, there are 2 solutions
What does 4^2n + 4^2n + 4^2n + 4^2n =
4 x 4^2n = 2^2 x 4^2n = 2^2 x 2^2(2n) = 2^2 x 2^4n) = 2^2 +4n
Typically the pattern with these "Units" digit problems have a period of ..
4,meaning every multiple of 4 repeats. Ex 7^1 will have the same units digit as 7^5, and 8^2 will have the same units digit as 8^6.
Multiplication is both ...
Communitive and Associative
What is the value of a base raised to a power of 0?
If the base does not equal zero, then it will equal 1. This is because a^1 /a^1 has an exponent of zero and a/a is 1.
When dealing with fractions, leaving a radical in the denominator is consider...
Improper. As a result you need to rationalize the fraction.
The term "Radicand" references...
The number under the radical sign
When multiplying and dividing radicals ...
Treat the whole numbers and radicals differently. By multiplying or dividing the whole numbers, and then multiplying or dividing the radicals.
The Conjugate
When rationalizing fractions with radicals involving addition or subtraction in the denominator, we do this by changing the addition or subtraction sign to its opposite.
The area of an equilateral triangle is ..
[Square root of 3 / 4] x (side)^2
Every pair of prime factors and every even power of a prime factor is ...
a perfect square.
Keep in mind that any even power of a square root can be written as ..
a power of a whole number. Ex: square root of 2 to the 48, can be written as (2^24)^2.
When factoring a polynomial, the greatest common factor can be a...
a variable, a variable raised to a power, a number, a number raised to a power, or a number raised to a variable. Ex: 3^n+5 + 3^n, factored equals 3^n(3^5 + 1)
When solving for the product of powers with the same base, be sure not to...
add the powers as well, but remember to multiply them instead.
Implementing the "Units digit strategies" can be helpful when solving simple...
addition and subtraction problems with "Powers". Ex: 33^2 + 37^2 , notice that the first term will produce a value with the units digit of "9". and the second term will produce a value with the units digit of 9 as well. So we are looking for answer with the units value of 9 + 9, or 8.
Similar to exponents, the square root do not distribute over ..
addition and subtraction.
Exponents cannot be distributed across..
addition and subtraction. (Common mistake) .There is no fixed pattern or law of exponents for the sum or difference of powers with the same base.
When problems solving, before you answer the question, be sure you understand what is being asked. This will give you a better idea of what the correct ...
answer should look like. This is important because the GRE will intentionally provide wrong answers for those who get the right answer to the wrong question.
Practice: When dealing with scientific notation, if you have a negative exponent, the value will get smaller...
as the exponent increases. Ex: 1.3 X 10^-3 is larger than 1.3 (x) 10^-4. This transformation is similar to dividing the first number by 10. Division always makes a number smaller. So when you are given a problem like 1.3(x) 10^-4 and you are told that it will be decreased by 1000, it s the same as saying 1.3 (x) 10^-4/10^3 or 1.3 (x) 10^-7.
Unlike the square root which has only a positive solution, the cubed root has ..
both a negative and positive solutions.
A mixed root is a value that ...
contains a whole number and a radical as well. Ex: [3 square root 2]
Something cubed can produce ...
either a positive or negative value
Square root, fourth root and sixth root are ...
even roots
Use the exponent as a way of calculating how many factors...
exist in the product value
An exponential equation is an equation where the variable is located within the ...
exponents
When solving any equation, always remember that the act of squaring produces ...
extraneous roots
Similar to quadratics equations, when dealing with equations and radicals, or equations where variables are under the radicals, you must be aware of ...
extraneous solutions, or solutions that make the equation false. There can be combination of solutions such as, one solution, two solutions, or no solutions.
When taking the square root of an expression involving addition or subtraction, you should always try to ...
factor out a value to transform the sum or difference expression into a product, where we can easily apply power properties. Ex. Square root of [ (4+2) + (4-1)], which can be changed to square root of [ 4(2-1)] = square root [(4)(1)] = square root (4) * square root (1) = 2
A base value raised to the 1/2 is the same as ..
finding the square root of that base value. 3^1/2 = square root of 3.
Remember that the "units" digit is the...
first place value to the left of the decimal point.
For positive numbers, the square root preserves the order of the ...
in equality. Meaning that 2 being less than 4, will maintain this relationship if we took the square root of 2 and the square root of 4.
You cannot apply any law of exponents if the bases..
involved are not the same
Even if there is addition or subtraction in the numerator, or multiplication with a radical in the denominator to rationalize the fraction ...
just multiple the numerator and denominator by the radical over itself.
When you take the square root of a number between one and zero, the value of the number gets...
larger.
When dealing with exponential equations, the first step is to...
make the bases equal( this will involve expressing both bases as powers of smaller bases). Second step is to set the exponents equal to each other. The third step is to solve for "x".
When problem solving on the GRE, you need to unpack as much information as possible by first asking what area of ...
math is the question testing you on. Next what are the trends in the answer choices. For instance do the answer choices have only numbers, numbers & variables, non integers. Identifying the trend will help you better understand how to go about solving the answer..
Multiplication is a simple and quicker way to do addition, similarly using exponents is a quicker way to do...
multiplication
Exponents can distribute across
multiplication and division, where (a/b)^2 = a^2/b^2, and (ab)^2 = (a^2)(b^2)
When simplifying and solving a problem involving roots, remember that all perfect squares removed from under the radical should be ...
multiplied together to create a mixed root. Example if the prompt asked you to solve for the square root of (12 x 32), we would remove the perfect squares of 2 and 4 out from under the radical, multiply them so that the final result would be 8 square root 6.
When rationalizing a fraction that has only the radical in the denominator position, all you have to do is ...
multiply that fraction by the denominator over itself.
Raising a power to a power results in ...
multiplying the exponents.
n^1 equals
n, there is only one factor of n. Also keep in mind that any number, factor or value that stands alone, has an exponent of 1.
A negative value that is raised to an odd power is ...
negative
Any odd root of a positive number is positive, similarly any odd root of a negative number is...
negative
Remember that a square root has a..
negative and positive solution.
You cannot take the square root of a...
negative number, this is undefined.
Remember that in the expression, x^2, "x" can be either
negative or positive. So when a prompt as you how many numerical integers squared can produce 144, 169 and 196, the answer is not 3 but 6 integer values.
Something squared will never produce a ...
negative value
Remember that x^2 = -4 has.
no solution, because nothing squared is negative.
Cubed root, fifth root and seventh roots are
odd roots
If the radical sign is printed on the test questions, then
only consider the positive solution.
When the product of a series of exponential powers emerge, this also emerges as well ...
patterns
When trying to estimate the value of the square root of an un-perfect square, simply take the square root of the nearest ...
perfect square before and after the target number, and that will let you know where that target number sits on the number line, and provide you with an idea of its value.
To over come or identify extraneous roots, simply ...
plug in the solutions and see which answer proves the equation true.
A negative value that is raised to an even power is...
positive
A square root only undoes a square, when the numbers are...
positive or is a zero value.
When the prompt provides you with two terms that are set equal to one another, where each term is raised to a fractional power, to eliminate the fractional power, just ...
raise each to the LCM of both fractional exponents. Ex 2x^(1/8) = 3x^(1/4), would be come [2x^(1/8)]^8 = [3x^(1/4)]^8, which would give you 2x = 3x^2.
When trying to figure out the order of a series of mixed roots, from smallest to largest, you can...
rewrite them as "entire roots". Ex: [3 square root 2], can be rewritten as [square root 9 (x) square root 2], which shows you are really dealing with the value square root 18.
When dealing with Q.C. always try to express both columns in the...
same form or use the same variable.
When the prompt states that expression [square root of the sum 7 and square root 13 ] can be written in the form square root of "a" plus square root of "b", then we can ...
set the two expressions equal to one another and solve. (Remember that when we are dealing with radicals, you always want to eliminate it by squaring both sides. So any time the prompt states that one expression can be written in the form of another, we can set the two expressions equal to one another.
You cannot simply add or subtract radicals, you must first...
simplify each radical and then add or subtract like-radicals.
When dealing with problems involving square roots, you should first always, always, always ...
simplify. If you see a square root then get rid of it by squaring both sides. If a fraction is in the exponent form, then find a way to turn it into an integer by multiplying it by its LCM. If a a radical is in the denominator form, use the conjugate rule to simplify. Etc....
When you take the square root of a number greater than one, the value of the number gets...
smaller
Whenever you encounter a problem where you are unsure how the exponent should relate or operate in the overall problem in terms of allowable properties, the easiest and quickest way to resolve this would be to ...
test the property you are questioning in a simple scenario using small numbers. This way you can quickly see what properties work and which ones are your misinterpretations.
Multiplying two powers with the same base means ...
that you can add the exponents.
Dividing two powers with the same base means ...
that you can subtract the exponents.
When solving for the product or division of two or more terms that contain variables raised to a power, use ..
the "associative property" to rearrange the order so that the coefficients are multiplied or divided together, and the like variables are multiplied or divided together. Then bring them back together at the end.
When the base value is negative with an absolute value less than one, by increasing the exponent....
the absolute value gets lower, yet the positive and negative sign alternates.
When the base value is negative with an absolute value greater than one, by increasing the exponent....
the absolute value of the number increases, yet the positive and negative sign alternates.
When dealing with exponents, as exponents get larger, its resultant value gets larger or smaller depending on...
the base value.
If a number is greater than 1, then the higher the root, the closer it will get to 1. Similarly if a number is less than one but greater than 0, the higher the root of that number...
the closer the number will get to one.
Whenever the prompt, asks you to figure out the possible values of a variable, given a certain situation, always think about...
the details of this situation that the test maker maybe trying to test you on. For example, if the prompt asks whether an even value raised to the x-1, could produce a positive or even value, the common answer would be "even".But remember the small detail that 1-1=0,and any value to the zero power is 1. So this expression will not always produce an odd number. Always be vigilant of the small details that the test maker is trying to trick you with.
When dealing with exponential equations, if the bases are the same on both sides of the equation then....
the exponents are the same or equal as well.
Assuming that all numbers are POSITIVE, the cubed root, square root, squaring, cubing , etc, preserves ...
the order of values. Ex if a>b then a^2>b^2
The radical sign automatically implies..
the positive solution.
When a fraction is an exponent of a base value, then the numerator is...
the power that the base value is raised to, and the denominator is the root that is being taken of the base value. Ex: 3^2/3 = the cubed root of 3^2.
When solving radical equations, equations where there is a variable under the radical sign, you can only square both sides when...
the radical is on a side by itself.
A base raised to a negative power is ...
the reciprocal of that same base to the positive power. In other words, A negative exponent can only become positive when it is moved to the denominator position.
The units digit of any product will only be influenced by..
the units digit of the two factors involved. You only need to consider "single digit products" when trying to solve unit digit problems.
When the base value is positive and larger than zero but less than one, by increasing the exponent....
the value decreases.
When the base value is positive and larger than one, by increasing the exponent....
the value increases as well
When dealing with power laws, to find the value of a base raised to 1, when you are given a value raised to a fraction, simply raise the original value to ...
the value of the denominator. Ex: k^1/4, can be written as (k^1/4)^4 = K^1= K
When a fraction is an exponent of a base value, if the denominator is odd ...
then the base can be negative or positive
B^n means..
there are "n" factors of "B" being multiplied together.
Its best to simpilfy radicals as you work through a problem, because simplified problems are easier..
to work with. If a large number is under the radical sign, just simplify it by factoring, and removing the perfect squares.
When a fraction is raised to a power, it may be easier to ..
turn the fraction to a decimal and then execute the power increase. Often times you maybe able to ignore the decimal point and simply multiply to find the value. Example (2/5)^3, is (0.40)^3 or (0.4)^3 which is (4 *4*4)=64 and move the decimal places three to the right, and you get 0.064
Distributing and factoring are..
two sides of the same coin.
Square root and the act of squaring..
undo each other, similar to addition and subtraction.
When bases are set equal to one another...
we can also equate the powers as well
Unlike the square root in which you cannot take the square root of a negative number, when dealing with cubed roots..
you can take the cubed/odd root of any number on the number line including both negative and positive.
When rationalizing fractions with radicals involving addition or subtraction in the denominator,
you multiply the numerator and the denominator by the denominator's conjugate.
If you engage in problem solving where a square root arises in the solving...
you should consider both the negative and positive solution.
