Perform the indicated operations and simplify the result. Leave your answer in factored form.

1) x/16 - 1/x all over 1 + 4/x

A) x+4/16
B) x-4/16
C) 16/x+4
D) 16/x-4
E) -16/x+4

I chose answer C. I saw this as x^2-16/ x+4.

Multiply the following using the FOIL method. Express your answer as a single polynomial in standard form.

2) (sqrt a + a)(sqrt a - a)

A) a - a^2
B) a + a^2
C) sqrt a - a^2
D) 2 sqrt a
E) a

I chose answer C.

Simplify the expression. Express answers so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0.

( 8 x^-1 / 3 y^-1)^-2

A) 9 y^2 / 64 x^2
B) 64 x^2 / 9 y^2
C) 64 y^2 / 9 x^2
D) 9 x^2 / 64 y^2
E) -9 x^2 /64 y^2

I chose answer A) 9 y^2 / 64 x^2

1. first multiply both top and bottom of your fraction by 16x/(16x) to get

(x^2 - 16)/(16x-64)

= (x+4)(x-4)/(16(x-4))
= (x+4)/16 which is probably what you meant when you typed a)

2. (√a + a)(√a - a) = a - a^2 , which is b)

3. ( 8 x^-1 / 3 y^-1)^-2
= [(8/x) / (3/y)]^-2
= [(8/x)(y/3)]^-2
= [(8y)/(3x)]^-2
= [(3x)/(8y)]^2
= 9x^2/(64y^2) or D)

SO for #2 the answer is a-a^2 or +a^2?

Perform the indicated operation and simplify the results. Leave your answer in factored form.

X+4 _ x-1
_X-6__x+5 =
X+5
(Simplify your answer. Type your answer in factored form. Use integers or fraction for any number in the expression

To solve the first problem, you will need to simplify the given expression step by step.

Step 1: Simplify the numerator and denominator separately.
Numerator: x/16 - 1/x
To add these fractions, you need a common denominator. The least common multiple (LCM) of 16 and x is 16x.
The first fraction becomes (x * x) / (16 * x) = x^2 / 16x.
The second fraction becomes (16 - 1) / (16 * x) = 15 / (16x).

Denominator: 1 + 4/x

Step 2: Combine the fractions.
Now, you have (x^2 / 16x - 15 / 16x) / (1 + 4/x).
To combine fractions, apply the rule of a common denominator.
The numerator becomes (x^2 - 15) / 16x.
The denominator remains the same.

Step 3: Factor the numerator.
The numerator can be factored as (x - sqrt(15))(x + sqrt(15)).

Step 4: Simplify the answer.
The simplified expression is ((x - sqrt(15))(x + sqrt(15))) / (16x).
So, the answer is C) 16/x+4.

For the second problem, you need to use the FOIL method to multiply the given expressions.

Step 1: Apply the FOIL method.
(sqrt(a) + a)(sqrt(a) - a) = (sqrt(a) * sqrt(a)) - (sqrt(a) * a) + (a * sqrt(a)) - (a * a)
= a - a^2 + a * sqrt(a) - a^2.

Step 2: Combine like terms.
The terms with a^2 can be combined to give -2a^2, and the remaining terms are a + a * sqrt(a).
So, the answer is A) a - a^2.

For the third problem, you need to simplify the given expression step by step.

Step 1: Simplify the expression inside the parentheses first.
8x^-1 / 3y^-1 = (8 / x) * (y / 3)

Step 2: Apply the negative exponent to make it positive.
(8 / x) * (y / 3) becomes (3y) / (8x).

Step 3: Apply the power rule for exponents.
(3y / 8x)^-2 = (8x / 3y)^2

Step 4: Simplify the expression by squaring each term.
(8x)^2 / (3y)^2 = 64x^2 / 9y^2

So, the answer is B) 64 x^2 / 9 y^2.