For any two numbers \(\displaystyle a ,b\), 

\(\displaystyle a^{2} - b ^{2} = (a+b) (a - b)\)

We can solve this problem most easily by taking advantage of this pattern, setting 

\(\displaystyle a = 333,335 , b =333,331\):

\(\displaystyle 333,335 ^{2} - 333,331 ^{2}\)

\(\displaystyle = \left (333,335 + 333,331 \right )\left (333,335 - 333,331 \right )\)

\(\displaystyle =666,666 \times 4\)

\(\displaystyle =2,666,664\)

Use the properties of exponents to raise the number to the fourth power:

\(\displaystyle \left (8 \times 10 ^{-3} \right ) ^{4}\)

\(\displaystyle = 8 ^{4} \times \left ( 10 ^{-3} \right ) ^{4}\)

\(\displaystyle = 4,096 \times 10 ^{-3 \times 4 }\)

\(\displaystyle = 4,096 \times 10 ^{-12 }\)

This is not in scientific notation, so adjust:

\(\displaystyle 4.096 \times 10^{3} \times 10 ^{-12 }\)

\(\displaystyle = 4.096 \times 10^{3 + (-12)}\)

\(\displaystyle = 4.096 \times 10^{-9}\)

For any two numbers \(\displaystyle a ,b\), 

\(\displaystyle a^{2} - b ^{2} = (a+b) (a - b)\)

We can solve this problem most easily by taking advantage of this pattern, setting 

\(\displaystyle a = 1,000,001 , b = 999,999\):

\(\displaystyle 1,000,001 ^{2} - 999,999^{2}\)

\(\displaystyle = \left ( 1,000,001+ 999,999 \right )\left ( 1,000,001- 999,999 \right )\)

\(\displaystyle = 2,000,000 \times 2 = 4,000,000\)

\(\displaystyle \left (6 \times 10 ^{4} \right ) ^{3}\)

\(\displaystyle = 6^{3} \times \left (10 ^{4} \right ) ^{3}\)

\(\displaystyle =216 \times 10 ^{4 \times 3 }\)

\(\displaystyle =216 \times 10 ^{12 }\)

This is not in scientific notation, so adjust:

\(\displaystyle 2.16 \times 10 ^{2 } \times 10 ^{12 }\)

\(\displaystyle =2.16 \times 10 ^{2 + 12 }\)

\(\displaystyle =2.16 \times 10 ^{14 }\)

\(\displaystyle 6.5*10^{4}\) is equal to \(\displaystyle 6.5*10,000\)

Move the decimal one place to the right for each number of the exponent with a base ten.

For example, \(\displaystyle 6.5 \ast10^{^{1}}=65\),  \(\displaystyle 6.5*10^{^{2}}=650\), etc.

By the Power of a Product Principle, 

\(\displaystyle (xy) ^{6} = x ^{6} \cdot y ^{6}\)

Also, by the Power of a Power Principle, 

\(\displaystyle (x^{3} ) ^{2} = x ^{3 \cdot 2 } = x ^{6}\)

Combining these ideas,

\(\displaystyle (xy) ^{6} = (x^{3} ) ^{2} \cdot y ^{6} = (-7 ) ^{2} \cdot 13 = 7 ^{2} \cdot 13 = 49 \cdot 13 = 637\)

\(\displaystyle \frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}} = \frac{ \left ( 30 -30\right )^{0}}{15^{0}}=\frac{ 0^{0}}{15^{0}}\)

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

Let's simplify both quantities first before we compare them. \(\displaystyle \left(\frac{1}{16}\right)^\frac{1}{2}\) becomes \(\displaystyle \sqrt{\frac{1}{16}}\)because the fractional exponent indicates a square root. We can simplify that by knowing that we can take the square roots of both the numerator and denominator, as shown by: \(\displaystyle \frac{\sqrt{1}}{\sqrt{16}}\). We can simplify further by taking the square roots (they're perfect squares) and get \(\displaystyle \frac{1}{4}\). Then, let's simplify Column B. To get rid of the negative exponent, we put the numerical expression on the denominator. There's still the fractional exponent at play, so we'll have a square root as well. It looks like this now: \(\displaystyle \frac{1}{\sqrt{\frac{1}{16}}}\). We already simplified \(\displaystyle \sqrt{\frac{1}{16}}\), so we can just plug in our answer, \(\displaystyle \frac{1}{4}\), into the denominator. Since we don't want a fraction in the denominator, we can multiply by the reciprocal of \(\displaystyle \frac{1}{4}\), which is 4 to get \(\displaystyle 1(4)\), which is just 4. Therefore, Column B is greater.

The reciprocal of \(\displaystyle 1.6 \times 10 ^{-7}\) is the quotient of 1 and the number;

\(\displaystyle \frac{1}{ 1.6 \times 10 ^{-7} }\)

\(\displaystyle = \frac{1 \times 10^{0}}{ 1.6 \times 10 ^{-7} }\)

\(\displaystyle = \frac{1 }{ 1.6 } \times \frac{ 10^{0}}{ 10 ^{-7} }\)

\(\displaystyle =0.625 \times 10^{0- (-7) }\)

\(\displaystyle =0.625 \times 10^{7 }\)

This is not in scientific notation, so adjust.

\(\displaystyle 6.25 \times 10^{-1 } \times 10^{7 }\)

\(\displaystyle = 6.25 \times 10^{-1+7 }\)

\(\displaystyle = 6.25 \times 10^{6 }\)

The reciprocal of \(\displaystyle 3.2 \times 10 ^{9}\) is the quotient of 1 and the number, or

\(\displaystyle \frac{1}{ 3.2 \times 10 ^{9} }\)

\(\displaystyle = \frac{1 \times 10^{0}}{ 3.2 \times 10 ^{9}}\)

\(\displaystyle = \frac{1 }{ 3.2 } \times \frac{ 10^{0}}{ 10 ^{9} }\)

\(\displaystyle =0.3125 \times 10 ^{-9}\)

This is not in scientific notation, so adjust:

\(\displaystyle 3.125 \times 10 ^{-1}\times 10 ^{-9}\)

\(\displaystyle =3.125 \times 10 ^{-1+ (-9)}\)

\(\displaystyle =3.125 \times 10 ^{-10}\)