The difference between the price to repair the car or buy a new one is $11,000. You divide this by the monthly savings, $220, giving you 50 months. 

To simplify this, we will want to use the correct order of operations. The mnemonic device PEMDAS is usually very helpful.

Parenthesis (1st)

Exponents (2nd)

Multiply, Divide (3rd)

Add, Subtract (4th)

PEMDAS tells us to evaluate parentheses first, then exponents. After exponents, we evaluate multiplication and division from left to right, and then we evaluate addition and subtraction from left to right.

Let's look at (xy)² – x((4x)(y)² – (4x)²) – 4²x²

We want to start with parentheses first. We will simplify the (xy)² by using the general rule of exponents, which states that (ab)^c = a^cb^c. Thus we can replace (xy)² with x²y².

x²y² – x((4x)(y)² – (4x)²) – 4²x²

When we have parentheses within parentheses, we want to move from the innermost parentheses to the outermost. This means we will want to simplify the expression (4x)(y)² – (4x)² first, which becomes 4xy² – 16x². We can now replace (4x)(y)² – (4x)² with 4xy² – 16x² .

x²y² – x(4xy² – 16x²) – 4²x²

In order to remove the last set of parentheses, we will need to distribute the x to  4xy² – 16x² . We will also make use of the property of exponents which states that a^ba^c = a^b+c.

x²y² – x(4xy²) – x(16x² ) – 4²x²
= x²y² – 4x²y² – 16x³ – 4²x²

We now have the parentheses out of the way. We must now move on to the exponents. Really, the only exponent we need to simplify is –4², which is equal to –16. Remember that –4² = –(4²), which is not the same as (–4)².

x²y² – 4x²y² – 16x³ – 16x²

Now, we want to use addition and subtraction. We need to add or subtract any like terms. The only like terms we have are x²y² and –4x²y². When we combine those, we get –3x²y²

–3x²y² – 16x³ – 16x²

The answer is –3x²y² – 16x³ – 16x² .

It's much easier to use factoring and canceling than it is to use long division for this problem. 9x² – 1 is a difference of squares. The difference of squares formula is a² – b² = (a + b)(a – b). So 9x² – 1 = (3x + 1)(3x – 1). Putting the numerator and denominator together, (9x² – 1) / (3x – 1) = (3x + 1)(3x – 1) / (3x – 1) = 3x + 1.

First, let's expand (a – b)² using the FOIL method. 

(a – b)(a – b) = a² – ab – ba + b² = a² – 2ab + b²

The value of a² + b² is already given to us. We can manipulate the equation ab = –8 to determine the value of –2ab. Multiply both sides of the equation by –2.

–2ab = 16

We will now substitute the values of –2ab and a² + b² to find the value of (a – b)².

(a – b)(a – b) = a² – 2ab + b² = 16 + 20 = 36

The answer is 36.

We are told that 4x – 8y = 18, but we want to know the value of 4y – 2x. In order to do this, we want to manipulate the expression 4x – 8y until it looks like 4y – 2x. We want to go from 4x to –2x. In order to do this, we could divide both sides of the equation by –2. If we did this, then we would also go from –8y to 4y. In short, we can find the value of 4y – 2x if we divide both sides of the equation 4x – 8y = 18 by negative two.

4x – 8y = 18

We can use the algebraic rule which states that , as long as c is nonzero. We will also use the fact that –a + b = b – a.

The answer is –9.

Using the multiplication property of logarithms we get. 

Knowing  we get

Assume there are x pennies.  Hence the number of nickels is . 

Now you have to set up an equation for the dollar value of the pennies and nickels which will be . 

Now solving for  results in (number of pennies).  Hence the number of nickels will be .

To solve this problem we use the logarithm rule

or

 can also be expressed as 

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x² + 2 = x² + x – 2 – x² + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0