Today is our topic of discussion Explanation of the Root .
Explanation of the Root
Definition: If nЄ N, n > 1 and a Є R, if there is such a Є R that ” this is called a n-th root of a. If n = 2 this root is called square root and for n = 3 it is called cube root.
Formula 7.
If a < 0, n€ N, n > 1 and n is odd then show that, √a = √|a|
Proof:
√a= -√|a| [ a<0]
= √(-1)|a| [. n is odd]
= √|a|
So, nx√|a|
Formula 8.
If a > 0, mЄ Z and nЄ N, n > 1, (nx√a)xm = √am
Proof:
Let, var and Vam: = = y
Then, “a and y” = am
or, y” = am = (x) = (xm)n
Since y > 0, x > 0, therefore considering the unique n-th root we get, y = xm
or, Vam = (a)
i.e, (a) = Vam
Formula 9.
If a > 0 and m = where m, pЄ Z and n, q € N, n > 1,q> 1
then, Vam Vap =
Proof:
Here, qm = pn
Let, Vam = x then, x” = am
or, (x) = (am)
or, nama = apm
or, (x)” = (a³)”
or, x = a” [considering the unique n-th root]
or, x = Vap
i.e, Vam = Vap
Corollary 1. If a > 0 and n, kЄ N, n > 1, then, vaak
Rational Fractional Exponents
Definition 5: If a Є R and n € N, n > 1, a = a when a > 0 or a < 0 and n are odd.
Remark: Exponent Rule (am)” = amn [Formula 6] =a”
If it is true in all cases, then (a)” = an = a¹ = a needs to happen, i.e n-th root of a needs to happen. So the definition mentioned above is explained to prevent ambiguity for more than one number of roots.
Remark: If a <0 and nЄ N, n > 1 odd then from formula 7 it is seen that a = √a = √√|a| = −|a|
In these cases using this formula, the value of a is determined.
Remark: If a is a rational number, then in most of the cases a will be a rational number too. In these cases, the approximate value of a is used.
Definition 6: If a > 0, m€ Z and nЄ N, n > 1 a = (a)m
Note: From definition 5 and 6 and formula 8 we see that, a = (a) = Vam where, a > 0, mЄ Z and, n Є N, n>1
m Р So, pЄ Z and q € Z,n> 1 if such that, = happens, then we can see from n q formula 9 that, a 2 = @”
Note: From the definitions of integer exponents and rational fractional exponents, we can get the explanation of a”, where a > 0 and r E Q. From the above discussion we see that if a > 0, then if r is divided into equal fractional value, the value of a” does not change.
Note: The rules explained in formula 6 is generally true for all exponents.
Formula 10.
If a > 0,b> 0 and r, s E Q
1) a”. a³ = a+s
2) απ =
3) (a”)” = a”s 4) (ab)” = a”b”
5) (1) 5)(c)= απ br
From repeated applications of (a) and (d) we see that,
Corollary 2. If
1) a > 0 and r1, T2, Tk EQ,
2) If a > 0, a₂ > 0,, an > 0 and r E Q then (a1 a₂ an) a・ aan.
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