Tabansi
2020-10-18
Answered

Radicals and Exponents Simplify the expression.

$\frac{{x}^{4}{\left(3x\right)}^{2}}{{x}^{3}}$

You can still ask an expert for help

Bentley Leach

Answered 2020-10-19
Author has **109** answers

Procedure used:

The definition of a Rational Exponent is,

For any rational exponent m/n in lowest, where m and n are integers and

If n is even, then we require that

This implies that

According to the Zero and Negative Exponents rule, we can see that if

From, the law of exponents, we know that

Calculation:

Using the Law of Exponents (1), (2) and (3), we get that

Answer:

The simplified form of the expression,

Jeffrey Jordon

Answered 2021-10-27
Author has **2313** answers

Answer is given below (on video)

asked 2022-02-08

How do you FOIL $(5-2\sqrt{2})}^{2$ ?

asked 2022-04-26

How to solve

$5-{\mathrm{log}}_{2}(x-3)={\mathrm{log}}_{2}(x+1)$

asked 2021-09-09

The equation $y=b{x}^{a}$ can be written as $\mathrm{log}y=\mathrm{log}b+a\mathrm{log}x$

asked 2022-05-21

Let $C$ be the algebraic curve defined by the modular polynomial ${\varphi}_{N}$ of order $N>1$ over the rational numbers, i.e.

$C:=\text{specm}(\mathbb{Q}[X,Y]/{\varphi}_{N}(X,Y)).$

The singularities of this curve can be removed and we obtain a nonsingular curve ${C}^{sn}$n, then, we can embed ${C}^{sn}$ into a complete non-singular curve $\overline{C}$.

In Milne's notes "Modular Functions and Modular Forms" it is written:

The coordinate functions $x$ and $y$ are rational functions on $\overline{C}$, they generate the field of rational functions on $\overline{C}$, and they satisfy the relation ${\varphi}_{N}(x,y)=0$.

I assume, by coordinate functions he means the functions $f(X),g(X)\in \mathbb{Q}[X]$ such that ${\varphi}_{N}(f(x),g(x))=0$ for all $x\in Q$. However, I don't understand why the field of rational functions on $\overline{C}$ is generated by these functions. Could someone explain this to me?

Thank you very much in advance!

$C:=\text{specm}(\mathbb{Q}[X,Y]/{\varphi}_{N}(X,Y)).$

The singularities of this curve can be removed and we obtain a nonsingular curve ${C}^{sn}$n, then, we can embed ${C}^{sn}$ into a complete non-singular curve $\overline{C}$.

In Milne's notes "Modular Functions and Modular Forms" it is written:

The coordinate functions $x$ and $y$ are rational functions on $\overline{C}$, they generate the field of rational functions on $\overline{C}$, and they satisfy the relation ${\varphi}_{N}(x,y)=0$.

I assume, by coordinate functions he means the functions $f(X),g(X)\in \mathbb{Q}[X]$ such that ${\varphi}_{N}(f(x),g(x))=0$ for all $x\in Q$. However, I don't understand why the field of rational functions on $\overline{C}$ is generated by these functions. Could someone explain this to me?

Thank you very much in advance!

asked 2022-07-16

Generalize logarithmic coincidences

After playing around with logarithms, I've found the following coincidences:

${\mathrm{log}}_{10}2\approx 0.3$, since ${2}^{10}\approx {10}^{3}$, and ${\mathrm{log}}_{10}5\approx 0.7$, since ${5}^{1000}\approx {10}^{700}$

I'm sure these are well known. I was just wondering if there was a method or algorithm to generalize these "coincidences" to any base or number. I only have a basic understanding of number theory and would like to learn more.

After playing around with logarithms, I've found the following coincidences:

${\mathrm{log}}_{10}2\approx 0.3$, since ${2}^{10}\approx {10}^{3}$, and ${\mathrm{log}}_{10}5\approx 0.7$, since ${5}^{1000}\approx {10}^{700}$

I'm sure these are well known. I was just wondering if there was a method or algorithm to generalize these "coincidences" to any base or number. I only have a basic understanding of number theory and would like to learn more.

asked 2021-06-17

Write two different rational functions whose graphs have the same end behavior as the graph of $$y=3{x}^{2}$$

asked 2021-12-06

Factor each polynomial

$2{x}^{2}+4x+1$