How To Solve: Exponents
Attached pdf of this Article as SPOILER at the top! Happy learning! Hi All,
I have recently uploaded a video on YouTube to discuss
Exponents in Detail:
Following is covered in the video
¤ Simplifying \((-1)^n\)
¤ Simplifying \((-k)^n\)
¤ Simplifying \(0^n\)
¤ Simplifying \(1^n\)
¤ Simplifying \((\frac{x}{y})^a\)
¤ Simplifying \(a^x * a^y\)
¤ Simplifying \(\frac{a^x }{ a^y}\)
¤ Simplifying \(x^a * y^a\)
¤ Simplifying \(x^{a^b}\)
¤ Simplifying \(a^{(-x)}\)
¤ Simplifying \(a^{(\frac{x}{y})}\)
¤ Adding exponents with same base and power
¤ \(x^n – y^n\) is ALWAYS divisible by x-y
¤ \(x^n – y^n\) is divisible by x+y when n is even
¤ \(x^n + y^n\) is divisible by x+y when n is odd
¤ \(x^n + y^n\) is NEVER divisible by x-y
Simplifying \((-1)^n\)\((-1)^n\)
= - 1 (for all odd values on n)
= + 1 (for all even values of n)
Simplifying \((-k)^n\)\((-k)^n\)
= 1 if n is 0
= +ve if n is even (except n = 0)
= -ve if n is odd
= 0 if k=0 and n≠0
= not defined if k=0 and n=0
Simplifying \(0^n\)\(0^n\) = 0 , for all n ≠ 0
Simplifying \(1^n\)\(1^n\) = 1 ( Always)
Simplifying \((\frac{x}{y})^a\)\((\frac{𝒙}{𝒚})^𝒂\)= \(\frac{𝒙^𝒂}{𝒚^𝒂}\)
Simplifying \(a^x * a^y\)If the base of two exponents is same and if we are multiplying the exponents, then we can keep the same base and add the powers.\(a^x * a^y = a^{( x + y )}\)
Simplifying \(\frac{a^x }{ a^y}\)If the base of two exponents is same and if we are dividing the exponents, then we can keep the same base and subtract the powers.\(\frac{𝒂^𝒙}{𝒂^𝒚} = a^{( x – y )}\)
Simplifying \(x^a * y^a\)If the power of two exponents is same and if we are multiplying the exponents, then we can multiply the bases and keep the same power\(x^a * y^a = (xy)^a\)
Simplifying \(x^{a^b}\)\(x^{a^b} = x^{b^a} = x^{ab}\)
Simplifying \(a^{(-x)}\)\(a^{-x} = \frac{1}{𝑎^𝑥}\)
Simplifying \(a^(\frac{x}{y})\)\(a^{𝑥/𝑦}= y√(a^x) = (y√a)^x\)
Adding exponents with same base and powerIf we are adding two or more exponents with the same power, then we can add them like normal variables\(x^a + x^a + x^a = 3 * x^a\)
\(x^n – y^n\) is ALWAYS divisible by x-yEx: If we take n = 2 then we have, \(x^n - y^n\) = \(x^2 - y^2\) = ( x - y ) * ( x + y) = divisible by x - y
\(x^n – y^n\) is divisible by x+y when n is EVENEx: If we take n = 1 then we have, \(x^n - y^n\) = \(x^1 - y^1\) = ( x - y ) => NOT divisible by x + y
Ex: If we take n = 2 then we have, \(x^n - y^n\) = \(x^2 - y^2\) = ( x - y ) * ( x + y) => divisible by x + y
\(x^n + y^n\) is divisible by x+y when n is ODDEx: If we take n = 2 then we have, \(x^n + y^n\) = \(x^2 + y^2\) => there is NO way in which we can express this as (x+y) * some other integer => NOT divisible by x + y
Ex: If we take n = 3 then we have, \(x^n + y^n\) = \(x^3 + y^3\) = ( x + y ) * ( \(x^2 - xy + y^2\)) => divisible by x + y
\(x^n + y^n\) is NEVER divisible by x-yEx: If we take n = 2 then we have, \(x^n + y^n\) = \(x^2 + y^2\) => there is NO way in which we can express this as (x-y) * some other integer => NOT divisible by x - y
Ex: If we take n = 3 then we have, \(x^n + y^n\) = \(x^3 + y^3\) = ( x + y ) * ( \(x^2 - xy + y^2\)) => there is NO way in which we can express this as (x-y) * some other integer => NOT divisible by x - y
Hope it helps!
Good Luck!