Algebra

Exponential Inequalities

Exponential Inequalities: Level 2 Challenges

         

7170or7071\Large \color{#3D99F6}{71^{70}} \quad \text{or} \quad \color{#D61F06}{ 70^{71} }

Which one of the two numbers above is greater?

True or False?

\quad If a,b,\color{#D61F06}{a},\color{#3D99F6}{b}, and c\color{#20A900}{c} are non-zero reals and a>b,\color{#D61F06}{a}>\color{#3D99F6}{b}, then ac>bc. \color{#D61F06}{a}^{\color{#20A900}{c}} > \color{#3D99F6}{b}^{\color{#20A900}{c}}.

How many positive integers nn are there such that 10nn10 10^n \leq n^{10}?

x(x2x2)<1\large |x|^{(x^2-x-2)} < 1

If the solution to the inequality above is x(A,B)x\in (A,B) , then find the value of A+BA+B.

Given three numbers such that 0<a<b<c<1 0 < a < b < c < 1, define

A=aabbcc,B=aabccb,C=abbcca. A = a^{a}b^{b}c^{c}, \quad B = a^{a}b^{c}c^{b} , \quad C = a^{b}b^{c}c^{a}.

How do the values of A,B,CA, B, C compare to each other?

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