Questions tagged [exponent]

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7 votes
1 answer

Rational Powers of Negative Numbers on Basic Calculators

Context: I teach at a small liberal arts school in the US. The students of interest in this question are generally in their first year. In courses like precalculus and calculus I like to use ...
  • 1,207
16 votes
11 answers

Natural occurrences of a to the (b to the c)?

Are there some natural contexts in which a double exponential occurs, $x$ to the ($y$ to the $z$): $$ x ^ {(y ^ z)} \;, \textrm{or} \;, a ^ {(b ^ c)} \; \textrm{?} $$ Of course one can contrive many ...
8 votes
4 answers

How should I convince a student who proved $1=-1$

One of my high school students who has ZERO knowledge on complex numbers and the modulus function has showed me the following algebra: $$(16)^{\frac{1}{2}}=(16)^{\frac{2}{4}}=((16)^2)^{\frac{1}{4}}=...
6 votes
0 answers

What books properly address the properties of $a^b$?

Many students think $\sqrt{a} \sqrt{b}=\sqrt{a\ b}$ $\sqrt{a^2}=a$ $\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}$ but none of the above are true when (a) and (b) are negative. To avoid such problems, ...
  • 241
8 votes
5 answers

Exponents with Negative Base; with or without Parentheses

How can I convincingly and mathematically explain the reason behind difference between $(-1)^2$ and $-1^2$? I used to add "negation" to the order of operations, in the same row as multiplication and ...
  • 2,323
7 votes
1 answer

Searching activities with "Find the error" strategy for learning maths

I read pages of the book "Elementary and Middle School Mathematics: Teaching Developmentally" by John A. Van de Walle, Late of Virginia Commonwealth University Karen S. Karp, Johns Hopkins University ...
  • 401