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

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

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.

\(\displaystyle \small 49+4+3=56\)