476: Squares and Cubes

How is this problem

related to this one from a few days back?

Is any square number times four another square number?

403: An Exponent Question

Considering the tweet below, which mental path do you think is easiest for beginning students?  (Analytical, numerical, graphical, algebraic?)

Would you give a different hint to beginning students than to advanced students?

#mathchat #SATPrep Will the average of 2^48 & 2^50 be <,>, or = 2^49?
Hint: Think graphically!
Now show the actual avg is 1.25×2^49.
— David Marain (@dmarain) March 18, 2015

365: Exponential Fun

If a and b are both positive integers, $a^{a - b}=10^6$,

list all possible ordered pairs (a,b).

What are a couple of different ways to approach this nugget?

What are the "obvious" answers that everyone will miss?

If a,b pos int's, a^(a-b)=10^6, list all possible ordered pairs (a,b)?
Ans:(10,4);(100,97);(1000,998); (1000000,999999)
— David Marain (@dmarain) January 3, 2015

278: Adding powers.

Take a moment to consider the rules and the methods that we use with powers and exponents.



2^x + 2^x + 2^x + 2^x = 2²⁰¹⁴

x = ?

What is the "Aha!" thought, the epiphany, in this problem?

#SATPREP #Patterns #Algebra 2^x + 2^x + 2^x + 2^x = 2^2014 x=? Answer? Solutions? Strategies?
— David Marain (@dmarain) September 20, 2014

226: Proof Without Words 5

from James Tanton Math Without Words Puzzle (solutions here: www.jamestanton.c...)

147: Exponents

If a, b are positive integers greater than 1 and b^a=2¹² then what is the largest possible value of b?

from D. Marain

128: Which is bigger?

We'll just let this one speak for itself ...

Quick, which infinite sum is greater?

1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 ...,
or
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 ...

— Matt Enlow (@CmonMattTHINK) April 29, 2014

114: Square Root Exponents

We all know that adding/subtracting exponents corresponds to multiplying/dividing the terms, like this:

$x^4 * x^7 = x^{11}$

$\dfrac{x^{14}}{x^9} = x^5$

Then negative exponents logically followed: $x^{-7} = \dfrac{x^2}{x^9}$

Then $\dfrac{x^3}{x^3} = x^0 = 1$ logically followed that. 

Additionally, multiplying/dividing the exponents relates to powers/roots

$ {x^4}^2 = x^{4*2} = x^8$

$\sqrt{x^6} = x^{6/2} = x^3$

So a fractional exponent means a radical, depending on the denominator of the exponent.

So here's my question:

What should we think about $x^{\sqrt{2}}$

How should we interpret that?

109: Powers of Two

If the tiles always pop up as 2s, how many moves has this player made?

62: Basketball.

There are 68 teams in the U.S. College Basketball Tournament.  The bottom eight play an extra round; the winners get the bottom seeds in the main tournament.  The main bracket has 64 teams in a single elimination tournament.

In the main bracket, how many games will be played, total?

If you just flipped a coin, what is the probability that you'd win them all?

Warren Buffet has offered $1 Billion Dollars to anyone who does pick all the games. What are the chances he will lose 1/60th of his money?

If you'd like to join the fun. with the rest of the Mathtwitterblogosphere.

39: zeroes AND exponents.

Jim Hays via email:

If anything to the power of 0 is 1, and 0 to any power is 0, then what is $0^0$?

30: Holy Moley Exponents

In the comments on Day 7b, More Exponents, Liz, on January 29th, said:

Wow, answer on Wolframalpha was pretty surprising!
Is it true for all numbers a and b, when a < b, then $a^b > b^a$?

Well? What do you say, Internet?

26: The Units Digit

Let's play "Spot the Pattern" !

What is the units digit of $3^{3^{3^{3}}}$ ?

16: Exponents and powers of ten.

Following on an earlier question from Day 7 ...

If I were to tell you that $2^{100} - 100^2 = 1,267,650,600,228,229,401,496,703,195,376$,
can you tell me what would change if I hadn't subtracted $100^2$ ?

What is $2^{100}$?

How do you know?

Just for the record, what -illion is that?

14: Negatives, Squares and the Mistaken Calculator

From Kenneth Tilton via email (tiltontec.com):

If -3 is a non-zero number and a non-zero number squared is positive,
why does $-3^2 = -9$?

I went to desmos.com and typed it in:

I know I've got my TI around here:

That can't be right.  Wolfram is always the definitive source, right?
http://www.wolframalpha.com/input/?i=-3^2

What the heck is going on?

11: More Exponents

Can you show if this is true?

$2^6 * 2^6 = 2^{11} + 2^{11}$

Can we generate more like this one ... simple and elegant?




How about one with 3 as a base?

Source:

This is a thing that made me look twice at it: 2^6 * 2^6 = 2^11 + 2^11.
— Kate Nowak (@k8nowak) January 21, 2014

8: More Exponents

From Dan Meyer (via email):

Which is bigger: $60^{63}$ or $63^{60}$?


The beauty of this problem is that both sides are beyond the capability of a TI-84 so a reduction of some kind is called for. wolframalpha.com has no difficulty but the answer is startling.


Find Dan on Twitter @ddmeyer, too.

7: Exponents.

What's bigger?  $2^{100} or {100}^2$ ?

@MathCurmudgeon Can I suggest one? What's bigger, 2^100 or 100^2?
— Max Goldstein (@dethornSTEM) January 21, 2014