Roots and Exponents Flashcards
sqrt(x^2) =
|x|
for even n, (x^n)^(1/n) =
|x| (always positive)
for odd n, (x^n)^(1/n) =
x (can be positive of negative)
sqrt(2) =
1.4
sqrt(3) =
1.7
sqrt(5) =
2.2
sqrt(6) =
2.4
sqrt(7) =
2.6
sqrt(8) =
2.8
radicals can be combined via multiplication if they the same..
index number ie. sqrt(5)*sqrt(2)=sqrt(10) since index of radical is 2
can radicals with different index numbers be combined?
no
what is the radicand
the expression under the radical
radicals can be added and subtracted if….
they are “like” (same root index and radicand)
Given the binomial a-sqrt(b), what is its conjugate
a+sqrt(b)
what is the product of a conjugate pair?
difference of squares
(a-b)(a+b)=a^2-b^2
T/F: (x/y)^(-n) = (y/x)^n
True
is it always necessary to check solutions of equations with radicals?
YES
In what cases does a^x=a^y not nessarily imply that x=y?`
if a=0,-1, 1
x^a * x^b = ?
x^(a+b)
For x != 0, (x^a)/(x^b) = ?
x^(a-b)
(x^a)^b =
x^ab (multiply exponents)
for different bases and like exponents
(x^a)(y^a) =
(xy)^a *almost acts like distributive property
*for different bases and like exponents
(x^a)/(y^a) =
(x/y)^a
Exponents can be distributed across (multiplication & division) or (addition & subtraction)?
multiplication & division
what is a convenient way to compare sizes of radicals that are practically impossible to compute?
raise each radical expression to the LCD of the root indices, and compare the sizes of the resulting expressions (which will just be exponent expressions). This works because of the following: for x, y, and m >0, then x>y iff x^m > y^m
Ex: 2^6 vs 3^4 — raise each side to (1/2) since 2 is the gcf of 4,6. Expressions reduce to 2^3 vs 3^2, which is easy to evaluate.. 8<9, and those 2^6 (which is 64) < 3^4 (which is 81)
2^(-2) can be re-expressed as?
1/(2^2)
When multiplying or dividing two numbers in scientific notation, can the coefficients and power of 10 be dealt with separately, and then combined at the end?
Yes, for example:
3.510^5 x 4010^6.. first multiply 3.540 to get 140, then tack on the product of the powers of 10 for a final product of 14010^11 (1.4*10^13 if answer needed in scientific notation)
sqrt(10,000) = ?
sqrt(81,000,000) = ?
sqrt(10,000) = 100
sqrt(81,000,000) = 9,000
When a perfect square ends with an even number of zeros, the square root of that perfect square will have exactly half as many trailing zeros as the original number
sqrt(0.0004) = ?
sqrt(0.000081) = ?
sqrt(0.0004) = 0.02
sqrt(0.000081) = 0.009
For a terminating decimal that is a perfect square, its square root will have exactly half as many digits as the original number (not half as many zeros!), thus a perfect square decimal has an even number of digits
cuberoot(27,000) =?
cuberoot(1,000) = ?
cuberoot(27,000) = 30
cuberoot(1,000) = 10
Since cube root of a perfect cube will have exactly 1/3 the number of zeros as the perfect cube
cuberoot(0.000027) = ?
cuberoot(0.000027) = 0.03
Since cube root of a perfect cube decimal will have exactly 1/3 the total number of decimal places (not just zeros decimal places!) as the original perfect cube decimal
(0.005)^2 =
0.000025
*total of 6 digits b/c of three initial -> 6 (double) total, then 5^2 = 25 (digits) and fill in remaining four as zeros
Protocol for squaring decimals is that the squared quantity will have a total of twice the number of existing decimal places, comprised of the square of the nonzero at the end with zeros as the remaining digits
(0.06)^3 = ?
(0.06)^3 = 0.000216
b/c 6^3 = 216, and b/c since 0.06 has two decimal places, (0.06)^3 will have 2*3 decimal places total
T/F: 2^n + 2^n = 2^(n+1)?
true - because of factoring
