For this problem we need to convert meters into kilometers and seconds into hours. Therefore we get,

$\displaystyle \frac{3.00\cdot10^8 m}{sec}\times \frac{60sec}{1min}\times \frac{60min}{1hour} \times\frac{.001km}{1m}$

Multiplying this out we get

$\displaystyle =1,080,000,000 = 1.08\cdot10^9$

100 miles = 528,000 feet. To put a number in scientific notation, we put a decimal point to the right of our first number, giving us 5.28. We then multiply by 10 to whatever power necessary to make our decimal equal the value we are looking for. For 5.28 to equal 528,000 we must multiply by 10^5.

Therefore, our final answer becomes:

$\displaystyle 5.28\times 10^5$

This question requires you to have an understanding of scientific notation. Begin by multiplying the two numbers:

$\displaystyle 0.0075 \cdot 0.0126 =0.0000945$

To use scientific notation, the number to the left of the decimal has to be between 1 and 10. In this case, we are looking to move the decimal place until we are left with 9 on the left of the decimal. Count the number of places that the decimal will have to move. In this case, it is five. Therefore:

$\displaystyle 9.45\cdot 10^{-5}$

Note: The notation is raised to a negative power because we moved the decimal from left to right.

Each year, you can calculate your interest by multiplying the principle ($\displaystyle \$5000$) by $\displaystyle 1.025$. For one year, this would be:

$\displaystyle 1.025*5000=5125$

For two years, it would be:

$\displaystyle 5125*1.025$, which is the same as $\displaystyle 1.025*1.025*5000$

Therefore, you can solve for a five year period by doing:

$\displaystyle 1.025^5*5000$

Using your calculator, you can expand the $\displaystyle 1.025^5$ into a series of multiplications. This gives you $\displaystyle 5657.041064453125$, which is closest to $\displaystyle \$5657$. 

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:

$\displaystyle \text{Final Balance}=\text{ Principal} \cdot \bigg(1+\frac{\text{Interest Rate}}{c}\bigg)^{(\text{Time})(c)}$

Plug in the values given:

$\displaystyle =10,000\cdot \bigg(1+\frac{.08}{4}\bigg)^{(1)(4)}$

$\displaystyle =10,000 \cdot (1.02)^{4}$

$\displaystyle =10,824.32$

Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:

$\displaystyle \text{Interest}=\text{Principal} \cdot \text{ Interest Rate} \cdot \text{ Time}$

$\displaystyle =5,000 \cdot (0.06)\cdot (1)$

$\displaystyle =300$

Add the two together, and we see that Jack makes a total of, $\displaystyle \$1124.32$ off of his investments.

It is easiest to break this down into steps. For each year, you will multiply by $\displaystyle 1.035$ to calculate the new value. Therefore, let's make a chart:

After year 1: $\displaystyle 7500*1.035=7762.5$; Total interest: $\displaystyle 262.5$

After year 2: $\displaystyle 7762.5*1.035=8034.1875$; Let us round this to $\displaystyle 8034.19$; Total interest: $\displaystyle 271.69$

After year 3: $\displaystyle 8034.19 * 1.035 = 8315.38665$; Let us round this to $\displaystyle 8315.39$; Total interest: $\displaystyle 281.2$

Thus, the positive difference of the interest from the last period and the interest from the first period is: $\displaystyle 281.2-271.69=9.51$

Let's pick numbers. For quantitative comparisons with exponents, it's good to try 0, a negative number, and a fraction.

0: 0² = 0, 0³ = 0, so the two quantities are equal.

–1: (–1)² = 1, (–1)³ = –1, so Quantity A is greater.

Already we have a contradiction so the answer cannot be determined.

We know that the expression must be negative. Therefore one or all of the terms x⁷, y⁸ and z¹⁰ must be negative; however, even powers always produce positive numbers, so y⁸ and z¹⁰ will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x⁷ must be negative, so x must be negative. Thus, the answer is x < 0.

First rewrite quantity B so that it has the same base as quantity A.

$\displaystyle 49^{^{-3}}$ can be rewriten as $\displaystyle (7^{^{2}})^{^{-3}}$, which is equivalent to $\displaystyle 7^{^{-6}}$.  

Now we can compare the two quantities.

$\displaystyle 7{^{-5}}$ is greater than $\displaystyle 7^{^{-6}}$.  

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

$\displaystyle 48^5^0+80^3^0=(2^4*3)^5^0+(2^4*5)^3^0$

Continuing the simplification:

$\displaystyle (2^4*3)^5^0+(2^4*5)^3^0=2^2^0^0*3^5^0+2^1^2^0*5^3^0$

Now, these factors have in common a $\displaystyle 2^1^2^0$. Factor this out:

$\displaystyle 2^1^2^0(2^8^0*3^5^0+5^3^0)$

Denominator

This is much simpler:

$\displaystyle 4^2^0=(2^2)^2^0=2^4^0$

Now, return to your fraction:

$\displaystyle \frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}$

Cancel out the common factors of $\displaystyle 2$:

$\displaystyle \frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}=2^8^0(2^8^0*3^5^0+5^3^0)=2^1^6^0*3^5^0+2^8^0*5^3^0$