$\displaystyle \frac{(-6)^{2} + (-6)^{3} + 6^{4}}{6} =\frac{ 36 + (-216) + 1,296 }{6} = \frac{1,116}{6} = 186$

$\displaystyle \frac{7^{2} + (-7)^{3} + (-7)^{4}}{7} =\frac{49 + (-343)+ 2401}{7} = \frac{2107}{7} = 301$

First, distribute the exponents to the numerators and denominators:

$\displaystyle \frac{3^{3}}{4^{3}}\times\frac{2^{2}}{9^{2}}$.

Then, rewrite the problem in expanded form:

$\displaystyle \frac{3\cdot3\cdot3}{4\cdot4\cdot4}\times\frac{2\cdot2}{9\cdot9}$

Next, rewrite the problem so that you only have prime factors in both numerators and denominators: 

$\displaystyle \frac{3\cdot3\cdot3}{2\cdot2\cdot2\cdot2\cdot2\cdot2}\times\frac{2\cdot2}{3\cdot3\cdot3\cdot3}$.

Last, simplify by "canceling out" like terms, leaving you with

$\displaystyle \frac{1}{2\cdot2\cdot2\cdot2}\times\frac{1}{3}$

and multiply across both numerators and denominators, which will give you $\displaystyle \frac{1}{48}$.

$\displaystyle \fn_cm When\; multiplying \; radicals, \; multiply \; the \; numbers \; under \; the \; radical\; and \; keep\; the\; radical.$

$\displaystyle \: \sqrt2 \times \sqrt18 = \sqrt36 = 6$

Use your power properties to multiply these numbers.

$\displaystyle \left (5 \times 10^{20} \right ) \left (8\times 10^{9} \right )$

$\displaystyle =\left (5 \cdot 8 \right )\times \left (10^{20} \cdot 10^{9} \right )$

$\displaystyle =40 \times 10^{20+9}$

$\displaystyle =40 \times 10^{29}$

This is not in scientific notation, so adjust as follows:

$\displaystyle 40 \times 10^{29}$

$\displaystyle =4 \times 10^{1} \times 10^{29}$

$\displaystyle =4 \times 10^{1+29}$

$\displaystyle =4 \times 10^{30}$

Use your power properties to multiply these numbers.

$\displaystyle \left (9.2 \times 10^{12} \right ) \left (1.5\times 10^{12} \right )$

$\displaystyle =\left (9.2 \cdot 1.5 \right )\times \left (10^{12} \cdot 10^{12} \right )$

$\displaystyle =13.8 \times 10^{12+12}$

$\displaystyle =13.8 \times 10^{24}$

This is not in scientific notation, so adjust as follows:

$\displaystyle 13.8 \times 10^{24}$

$\displaystyle =1.38 \times 10^{1} \times 10^{24}$

$\displaystyle =1.38 \times 10^{1+24}$

$\displaystyle =1.38 \times 10^{25}$

$\displaystyle 243_{\textrm{five}}$

$\displaystyle = 2 \cdot 5^{2} + 4 \cdot 5^{1 } + 3$

$\displaystyle = 2 \cdot 25 + 4 \cdot 5 + 3$

$\displaystyle = 50 +20+3 = 73$

$\displaystyle 231_{\textrm{six}}$

$\displaystyle = 2 \cdot 6^{2} + 3 \cdot 6^{1} + 1$

$\displaystyle = 2 \cdot 36 + 3 \cdot 6 + 1$

$\displaystyle = 72+ 18+ 1 =91$

In this problem, you need to use the proper order of operations to find the correct answer: PEMDAS (Parentheses, Exponents, Multiplication-Division, Addition-Subtraction).

Exponents show up first in this problem, so the first step is to simplify:

$\displaystyle 10^2 = 100$

This results in:

$\displaystyle 4+5*100-5$

The next step is the multiplication:

$\displaystyle 5*100=500$

This results in:

$\displaystyle 4+500-5$

The next step is to do the remaining addition and subtraction from left to right.

$\displaystyle 4+500-5=499$

499 is your final answer.

Follow the correct order of operations: parenthenses, exponents, multiplication, division, addition, subtraction.

$\displaystyle \small (3+4)^2+(\frac{3+5}{2})+6\div 2$

First, evaluate any terms in parenthesis.

$\displaystyle (7)^2+(\frac{8}{2})+6\div 2$

$\displaystyle 7^2+4+6\div 2$

Next, evaluate the exponent.

$\displaystyle \small 49+4+6\div2$

Divide.

$\displaystyle \small 49+4+3$

Finally, add.

$\displaystyle \small 49+4+3=56$