Question

solve this problem
$( {a}^{2} + 3 {b}^{2} ) \times ( {a}^{2} - \frac{2}{3} {b}^{2} )$
​

Answer 1

Subscripts and Superscripts

Subscripts and superscripts (such as exponents) can be made using the underscore _ and carat ^ symbols respectively.

Symbol Command Symbol Command

$2^{2}$ 2^2 $\textstyle a_i$ a_i

$\textstyle 2^{23}$ 2^{23} $\textstyle n_{i-1}$ n_{i-1}

$a^{i+1}_3$ a^{i+1}_3 $x^{3^2}$ x^{3^2}

$2^{a_i}$ 2^{a_i} $2^a_i$ 2^a_i

Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than one character, you must surround with curly braces. For example, x^10 produces $x^10$, while x^{10} produces $x^{10}$.

Math Commands

Here are some commonly used math commands in LaTeX:

Fractions

Symbol Command

$\frac {1}{2}$ \frac{1}{2} or \frac12

$\frac{2}{x+2}$ \frac{2}{x+2}

$\frac{1+\frac{1}{x}}{3x + 2}$ \frac{1+\frac{1}{x}}{3x + 2}

Notice that with fractions with a 1-digit numerator and a 1-digit denominator, we can simply group the numerator and the denominator together as one number. However, for fractions with either a numerator or a denominator that requires more than one character (or if the numerator starts with a letter), you need to surround everything in curly brackets.

Use \cfrac for continued fractions.

Expression Command

$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}$ \cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}

Radicals

Symbol Command

$\sqrt{3}$ \sqrt{3}

$\sqrt{x+y}$ \sqrt{x+y}

$\sqrt{x+\frac{1}{2}}$ \sqrt{x+\frac{1}{2}}

$\sqrt[3]{3}$ \sqrt[3]{3}

$\sqrt[n]{x}$ \sqrt[n]{x}

Sums, Products, Limits and Logarithms

Use the commands \sum, \prod, \lim, and \log respectively. To denote lower and upper bounds, or the base of the logarithm, use _ and ^ in the same way they are used for subscripts and superscripts. (Lower and upper bounds for integrals work the same way, as you'll see in the calculus section)

Symbol Command

$\textstyle \sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\textstyle \prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\textstyle \lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

$\textstyle \lim\limits_{x\to\infty}\frac{1}{x}$ \lim\limits_{x\to\infty}\frac{1}{x}

$\textstyle \log_n n^2$ \log_n n^2

Some of these are prettier in display mode:

Symbol Command

$\textstyle \cos^2 x +\sin^2 x = 1$ \cos^2 x +\sin^2 x = 1

$\cos 90^\circ = 0$ \cos 90^\circ = 0

Calculus

Below are examples of calculus expressions rendered in LaTeX. Most of these commands have been introduced before. Notice how definite integrals are rendered (and the difference between regular math and display mode for definite integrals). The \, in the integrals makes a small space before the dx.

Symbol Command

$\frac{d}{dx}\left(x^2\right) = 2x$ \frac{d}{dx}\left(x^2\right) = 2x

$\int 2x\,dx = x^2+C$ \int 2x\,dx = x^2+C

$\int^5_1 2x\,dx = 24$ \int^5_1 2x\,dx = 24

$\int^5_1 2x\,dx = 24$ \int^5_1 2x\,dx = 24

$\frac{\partial^2U}{\partial x^2} + \frac{\partial^2U}{\partial y^2}$ \frac{\partial^2U}{\partial x^2} + \frac{\partial^2U}{\partial y^2}

$\frac{1}{4\pi}\oint_\Sigma\frac{1}{r}\frac{\partial U}{\partial n} ds$ \frac{1}{4\pi}\oint_\Sigma\frac{1}{r}\frac{\partial U}{\partial n} ds

Overline and Underline

Symbol Command

$\overline{a+bi}$ \overline{a+bi}

$\underline{747}$ \underline{747}

LaTeX

Other Functions

Symbol Command Symbol Command Symbol Command

$\arg$ \arg $\textstyle\deg$ \deg $\textstyle\det$ \det

$\dim$ \dim $\textstyle\exp$ \exp $\textstyle\gcd$ \gcd

$\hom$ \hom $\inf$ \inf $\ker$ \ker

$\textstyle\lg$ \lg $\liminf$ \liminf $\limsup$ \limsup

$\textstyle\max$ \max $\textstyle\min$ \min $\Pr$ \Pr

$\sup$ \sup $\smiley$ \smiley

Some of these commands take subscripts in the same way sums, products, and logarithms do. Some render differently in display mode and regular math mode.

Symbol Command Symbol Command Symbol Command

$\dim_x$ \dim_x $\textstyle\gcd_x$ \gcd_x $\inf_x$ \inf_x

$\liminf_x$ \liminf_x $\limsup_x$ \limsup_x $\textstyle\max_x$ \max_x

$\textstyle\min_x$ \min_x $\Pr_x$ \Pr_x $\sup_x$ \sup_x

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Answer By shreyas