I would appreciate it sooo much Mail 12 Which number is bigg

Mail 12) Which number is bigger (1010)10 or 101 Explain. 13) Give an example of three irrational numbers r, y, and z such that ()* is rational. 14) Classify the following as polynomials, rational functions, or neither. (a) f(x) = x4-3x-10 (b) f(x) = x4-3d-10 (c) f(x) = (e) f(x) = 18 (d) f(x)- 35 3+1 15) Give an example of polynomials p and q of degree 5 such that deg(p+q) does not equal 5. Solve the following:

Solution

12) (10)^{(10)^(10)} is greater than (10¹⁰)¹⁰

since (a^m)ⁿ = a^mn ==> (10¹⁰)¹⁰ = 10^10*10

==> (10¹⁰)¹⁰ = 10¹⁰⁰

where as (10)^{(10^10)} = 10^10000000000

Hence (10)^{(10^10)} is greater than (10¹⁰)¹⁰

13) x ,y , z must be irrational and (x^y)^z must be rational

==> x = 2 , y = 3 , z = 12

==> (x^y)^z = (2³)¹²

we know that (a^m)ⁿ = a^mn

==> (2³)¹² = (2³*¹² )

==> 2³⁶

==> (2)⁶

==> (2^1/2)⁶

==> 2^(1/2)*6

==> 2³

==> 8

Hence (x^y)^z = (2³)¹² = 8 is a rational number

15) P and Q are 5^th degree polynomials

let P(x) = x⁵ + 2x⁴ + 6x³ + 2x² + 8x + 5

let Q(x) = -x⁵ + 3x⁴ + 7x³ + 6x² + 5x + 2

P + Q = x⁵ + 2x⁴ + 6x³ + 2x² + 8x + 5-x⁵ + 3x⁴ + 7x³ + 6x² + 5x + 2

==> P + Q = 5x⁴ + 13x³ + 8x² + 13x + 7

degree of P + Q = 4