Hi, I would like someone to explain to me the process to get to the answer of these questions. (The answers are in the back of the book, but I don't understand how to get them myself):

QUESTION ONE
Warren has 40 coins (all nickels, dimes, and quarters) worth $4.05. He has 7 more nickels than dimes. How many quarters does Warren have?

QUESTION TWO
Find and simplify the sum of a^2(ab^3)^2, b^2(a^2b^2)^2
("^2" means "exponent of two" or "to the second power")

QUESTION THREE
Find and simplify the sum of
a(-ab^2)^3 + (2a^2b^3)^2

QUESTION FOUR
Find and simplify the product of
a(-ab^2)^3 x (2a^2b^3)^2

QUESTION FIVE
Show that 16^x times (4^x)^2 = (2^x)^8

I would really, really appreciate any help for any of the questions. I am trying so hard but I just don't get it!

1. Are you sure that is all the data you have? I can't solve without some relationship with quarters.

N = D + 7

25Q + 10D + 5N = 405

25Q + 10D + 5(D+7) = 405

2. (ab^3)^2 = a^2(b^6), assuming that only the b term is cubed. Power to a power is simplified by multiplying.

That should help you with the other problems too.

I'd be happy to help you understand how to find the answers to these questions! Let's go through each question step by step.

QUESTION ONE:
To find the number of quarters Warren has, we need to set up a system of equations based on the given information.

Let's assign variables to the unknowns:
Let "n" represent the number of nickels,
"d" represent the number of dimes, and
"q" represent the number of quarters.

We know that Warren has 7 more nickels than dimes, so we can write the equation: n = d + 7.

We also know that there are a total of 40 coins, so the equation for the total number of coins is: n + d + q = 40.

We know the value of each coin: 0.05 for a nickel, 0.10 for a dime, and 0.25 for a quarter.

The total value of the coins is given as $4.05, so we can write the equation: 0.05n + 0.10d + 0.25q = 4.05.

Now we have a system of three equations. We can solve it to find the values of n, d, and q.

QUESTION TWO:
To simplify the expression a^2(ab^3)^2 + b^2(a^2b^2)^2, we need to use the rules of exponents and perform the necessary calculations.

Let's simplify each term separately:
a^2(ab^3)^2 = a^2(a^2b^6) = a^(2+2)b^6 = a^4b^6

b^2(a^2b^2)^2 = b^2(a^4b^4) = (b^2)(a^4)(b^4) = a^4b^6

Now that we have simplified both terms, we can add them together: a^4b^6 + a^4b^6 = 2a^4b^6.

QUESTION THREE:
To find and simplify the sum of a(-ab^2)^3 + (2a^2b^3)^2, we need to apply the rules of exponents and perform the calculations.

Let's simplify each term separately:
a(-ab^2)^3 = a(-a^3b^6) = -a^(1+3)b^6 = -a^4b^6

(2a^2b^3)^2 = (2^2)(a^2)^2(b^3)^2 = 4a^4b^6

Now, we can add the simplified terms together: -a^4b^6 + 4a^4b^6 = 3a^4b^6.

QUESTION FOUR:
To find and simplify the product of a(-ab^2)^3 x (2a^2b^3)^2, we need to apply the rules of exponents and perform the calculations.

Let's simplify each term separately:
a(-ab^2)^3 = a(-a^3b^6) = -a^(1+3)b^6 = -a^4b^6

(2a^2b^3)^2 = (2^2)(a^2)^2(b^3)^2 = 4a^4b^6

Now, we can multiply the simplified terms together: -a^4b^6 x 4a^4b^6 = -4a^8b^12.

QUESTION FIVE:
To show that 16^x times (4^x)^2 = (2^x)^8, we need to simplify both sides of the equation using the properties of exponents.

Let's simplify each side of the equation:
16^x = (2^4)^x = 2^(4x)

(4^x)^2 = (2^2)^2x = 2^(4x)

(2^x)^8 = 2^(x * 8) = 2^(8x)

Now, we can compare the two sides of the equation:
2^(4x) x 2^(4x) = 2^(4x + 4x) = 2^(8x)

Therefore, 16^x times (4^x)^2 is equal to (2^x)^8.

I hope this helps you understand how to find the answers for these questions! Let me know if you have any further questions or need additional explanations.