Algebra

Question 1

arq= a(r+s)
(ar)s = ars
(ab)r = arbr
\(\left(\frac{a}{b}\right)^r\)\(\frac{a^r}{b^r}\) = arb-r
\(\left(\frac{a^r}{a^s}\right)\)= a(r-s)

Question 2

Knowing the surface area of a sphere with radius r is 4\(\Pi\)r², quadrupling the radius of this sphere equates to giving it a radius of 4r. Consequently, the surface area of the sphere will be of  4\(\Pi\)4r² = 16\(\Pi\)r².       
If the radius of the sphere quadrupled, the surface area would increase by a factor of 4.
Similarly, if the radius of the sphere increased by a factor of x, the surface area would increase by a factor of x. The surface area would be equal to  4\(\Pi\)xr².   

Question 3

The general formula for the calculation of interest is the following : A = P (1 + rt), where:
A = total accrued amount
P = principal amoung
r= yearly interest rate
t = unit of time (years)
1,000,000 = 3,000 (1 + 40r)
1,000,000 = 3,000 + 120,000r
997,000 = 120,000r
r = \(\frac{997}{120}\)
r = 8.308
Therefore, the yearly interest rate should be of 830.8% in order to obtain $1 million in 40 years, starting with $3,000 today.

Logic, proofs and set theory

Question 1