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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 power
If 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-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 EVEN
Ex: 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 ODD
Ex: 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-y
Ex: 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