Question

$(3 + \sqrt{3) \: (3 - \sqrt{3)} }$
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Answer 1

Answer:

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

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

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

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

Note that we can use sums, products, and logarithms without _ or ^ modifiers.

Symbol Command

$\sum\frac{1}{i}$ \sum\frac{1}{i}

$\frac{n}{n-1}$ \frac{n}{n-1}

$\textstyle \log n^2$ \log n^2

$\textstyle \ln e$ \ln e

Mods

Symbol Command

$9\equiv 3 \bmod{6}$ 9\equiv 3 \bmod{6}

$9\equiv 3 \pmod{6}$ 9\equiv 3 \pmod{6}

$9\equiv 3 \mod{6}$ 9\equiv 3 \mod{6}

$9\equiv 3\pod{6}$ 9\equiv 3 \pod{6}

Combinations

Symbol Command

$\scriptstyle\binom{1}{1}$ \binom{1}{1}

$\scriptstyle\binom{n-1}{r-1}$ \binom{n-1}{r-1}

These often look better in display mode:

Symbol Command

$\dbinom{9}{3}$ \dbinom{9}{3}

$\dbinom{n-1}{r-1}$ \dbinom{n-1}{r-1}

Trigonometric Functions

Step-by-step explanation:

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Answer 2

Answer:

In chemistry, there are three definitions in common use of the word base, known as Arrhenius bases, Brønsted bases, and Lewis bases. All definitions agree that bases are substances which react with acids as originally proposed by G.-F. Rouelle in the mid-18th century.

The range goes from 0 - 14, with 7 being neutral. pHs of less than 7 indicate acidity, whereas a pH of greater than 7 indicates a base.

A base is a substance that accepts hydrogen ions. When a base is dissolved in water, the balance between hydrogen ions and hydroxide ions shifts the opposite way. Because the base "soaks up" hydrogen ions, the result is a solution with more hydroxide ions than hydrogen ions. This kind of solution is alkaline.

The lower, basic, or fundamental part of an object, organ, or substance. Anatomic nomenclature for the base of a structure or organ, or the part opposite to or distinguished from the apex.

Step-by-step explanation:

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