query: Prove that $|A^{B×C}| = |(A^B)^C|$
positive_passages.docid: 31173
positive_passages.text: How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
negative_passages.docid: 105367
negative_passages.text: Proving $||\vec{a}+\vec{b}|| = ||\vec{a}-\vec{b}|| \iff \vec{a} \perp \vec{b}$
negative_passages.docid: 14221
negative_passages.text: How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
negative_passages.docid: 111295
negative_passages.text: Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 65923
negative_passages.text: How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
negative_passages.docid: 63535
negative_passages.text: Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
negative_passages.docid: 101519
negative_passages.text: What is the condition for example $(number^c)^b=number^{cb}$ to be true?
negative_passages.docid: 118981
negative_passages.text: Proving $\sqrt{ab} = \sqrt a\sqrt b$
negative_passages.docid: 29462
negative_passages.text: Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
negative_passages.docid: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
negative_passages.docid: 95852
negative_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$
negative_passages.docid: 76049
negative_passages.text: Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
negative_passages.docid: 30770
negative_passages.text: Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
negative_passages.docid: 17176
negative_passages.text: Proof that $x^n \mod b = (x \mod b)^n$