Roots and Exponents
What is the principal square root?
the positive square root. When the radical symbol (√) is used, we only consider the principal square root
What are two things you need to remember when doing math problems?
1) factor, factor, factor 2) prime numbers
How do you rationalize the denominator?
Multiply the entire fraction by the radical value of 1 3+√5 ---- = 3√5/5 + 1 √5
What are the first eleven non-negative perfect cubes?
0 1 8 = 2 27 = 3 64 = 4 125 = 5 216 = 6 343 = 7 512 = 8 729 = 9 1000 = 10
What is a special rule to know when adding like bases with equal exponents?
2^n + 2^n = 2^(n+1)
What is the rule when dividing radicals?
2√100 ------ = √4 = 2 2√25
What is the rule when adding/subtracting radicals?
All of the operations under the radical must take place prior to taking the root
How can we convert numbers to scientific notation?
Also note: x^-n (means move the decimal to the left) x^n (means move the decimal to the right)
How do you remove the radicals in an equation?
By raising each expression to the LCD not 1/60, but just 60
How do you handle multiplying and dividing non-radicals and radicals?
Divide/multiply radicals by radicals, and non-radicals by non-radicals
What is the rule for "if the bases are equal, consider that the exponents may be equal"?
Ex., 2^x + 2^y = 2^10 (it must be true that x+y = 10, however, 1^10=1^20 but 10 doesn't equal 20
How do you add/subtract like bases?
Factor out the common factor x^4 + x^4 + x^4 + x^4 = x^4 (1+1+1+1) = x^4(4) = 4x^4 Not x^16
What is the conjugate rule to simplify the denominator if it contains a radical?
Multiply the fraction by the conjugate of the denominator divided by itself
What can you do with the coefficients in scientific notation?
When multiplying or dividing numbers in scientific notation (or almost in scientific notation), the coefficients can be multiplied or divided separately from the powers of ten
How do you add/subtract fractions that contain exponents?
You need to get the LCD To get the LCD of denominators that have exponents, we raise each unique base to the highest power that base has within the denominators
What is a like radical? What is the rule?
add or subtract like radicals only. Two or more radicals are like radicals if they both have the same root index and the same radicand (expression under the radical) 5√4 + 4√4 --> the common radicand of 4, so it becomes 9√4
If n is even, and x is nonnegative, then is the output negative or positive?
always positive
What to do if the bases are not the same?
attempt to make them the same through factorization
What is a conjugate pair? What is a binomial?
conjugate pair: a pair of binomials with identical terms but parting opposite arithmetic operators in the middle of these similar terms Ex., a-b <> a+b binomial: an algebraic expression of the sum or the difference of two terms (2x + 3)
What is the cube root of a perf/non-perfect cube?
cube root of a perfect cube = integer cube root of a non-perfect cube = non-integer
What are the cubed roots of non-perfect single-digit numbers?
cubed of 2 = ~1.3 cubed of 3 = ~1.4 cubed of 4 = ~1.6 cubed of 5 = ~1.7 cubed of 6 = ~1.8 cubed of 7 = ~1.9 cubed of 9 = ~2.1
How many zeroes does a cube root of a perfect cube integer/decimal have?
exactly one-third the number of zeroes/decimal places to the right of the final nonzero digit ex., 3√1,000,000 = 100 3√1,000 = 10 3√0.000027 = 0.03
How do you rewrite a fractional base that is raised to a negative exponent?
flip the fraction and make the exponent positive
What is the rule when comparing radicals and exponents?
for radicals, raise it to the LCD for exponents, raise it to the reciprocal of the GCF
What are the fourth roots of non-perfect single-digit numbers?
fourth root of 2 = ~1.2 fourth root of 3 = ~1.3 fourth root of 4 = ~1.4 fourth root of 5 = ~1.5 fourth root of 6 = ~1.6 fourth root of 7 = ~1.6 fourth root of 8 = ~1.7 fourth root of 9 = ~1.7
What is the output of a cubed (odd) root if n is even, what about odd?
if n is even, then it's the absolute value of x is n is odd, then it's x
What is the square root of a perfect square? what about a non-perfect square?
perfect square: whole number non-perfect square: not a whole number
What to do if you're dividing different bases that share a common exponent?
keep the common exponent and divide the bases
What to do if you're multiplying different bases that share a common exponent?
keep the common exponent and multiply the bases
How do you multiply like bases?
keep the like base and add the exponents Note: only do this when multiplying like bases, not adding!!
How do you divide like bases?
keep the like base and subtract the exponents
What prime numbers is the positive perfect cube made up of?
multiples of 3
How do you square a binomial?
multiply the binomial by itself using the FOIL process binomial: addition or subtraction of two terms
What is the power to a power rule?
multiply those exponents
What is the rule on radicals being expressed in the exponential form?
note: when taking the square root of an exponent, we cut the exponent in half since √x = x^1/2
What is a radicand? Can you combine radicands with different index numbers?
radicand: The value inside the radical symbol. no, but you can combine radicands with the same index number (see image) Ex., √5 * √8 = √40, which = √4 * √10 = 2√10
Review this example of solving equations with the variable raised to an even power
remember that when solving for x in this way, you are looking for the absolute value of the equation so if it's (x+6)^2 = 49 it becomes |x+6| = 7, so you're looking for values that would equate to both +7 and -7
How to handle a base raised to a negative exponent?
take the reciprocal of the base and exponent while turning the negative exponent into a positive one
What is the sq root of a number squared equal to?
the absolute value of that number
What is the output of a squared binomial or binomial raised to an even power under the root sign?
the absolute value of the binomial
If there are multiple variables in parentheses raised to a power, how does the distribution work?
the exponent is applied to every constituent in the parentheses
What is a radical's index?
the number found on the top left of the radical Note: the index of a square root is 2
What happens if an integer/decimal ending with zeros is a perfect square? (and has a finite number of decimal places)
then its square root will have exactly half the number of zeroes/decimal places ex., √10,000 = 100 √81,000,000 = 9,000 √0.0004 = 0.02
What is a rule to remember when two numbers share the same base but have exponents that differ by as little as 1?
these values can be vastly different from each other 9^8 = 43,046,721 9^7 = 4,782,969 Ex., The GMAT will test by asking, "Which of the following is closest to the value of 20^12 - 6^4?" (typically be closer to 20^12)
What is the form of scientific notation?
when a positive number n is written in scientific notation, it's written int he form of a x 10^b where a is a number between 1 and 10 (including 1 but excluding 10) and b is an integer
What is considered a simplified radical expression?
when the radical is removed from the denominator
Can we take the cubed root of a negative number?
yes cubed root of -8 = -2 (since -2*-2*-2 = -8)
What are the square roots of non-perfect single-digit numbers?
√2 = ~1.4 √3 = ~1.7 √5 = ~2.2 √6 = ~2.4 √7 = ~2.6 √8 = ~2.8