A circle with diameter 12 inches has a radius of 6 inches. The area of a circle is A = πr², where r is the radius. Thus, the area of this circle is A = π(6²)= 36π square inches.

If a^b = m and c^2b = n, then mn = (a^b)(c^2b). Because a and c are not necessarily the same number, you cannot simply add the exponents. The only legitimate answer choice that further manipulates the expression (a^b)(c^2b) factors out the b exponent and places it outside a set of parentheses. Thus(a^b)(c^2b) = (a^bc^2b) = (ac²)^b.

A good way to go about solving this problem is to think of the tests as each having a value of 100 points, making the total of all 3 tests worth 300 points. In order to average 87% overall, Sulema would then need to score 87% of the 300 points, or 300(0.87), or 261 points. By scoring 88% and 79% on her previous tests, she has already obtained 88 + 79, or 167 points. Thus Sulema must score 261 − 167, or 94 points on the third test, which is equivalent to 94%.

For the equation 3x² − 2x − 21 = 0, all solutions of x can be found by either factoring or by using the quadratic formula \(\left [ x=\frac{(-b\pm \sqrt{b^{2}− 4ac})}{2a}for quadratic equations ax^{2} + bx + c = 0 \right ]\). In this case, since the final term is negative, you know that when this equation is factored, one sign will be positive and the other will be negative. Knowing that the factors of 21 are 1 and 21 and 3 and 7, you can guess and check your answers with relative ease. In this case, the correct answer is (3x +7)(x −3) = 0. Solve for x: x = −⁷⁄₃ or x = 3.

In a rectangle with length l which is 5 inches longer than the width w, the relation between the sides is l = w + 5. Since perimeter, which is the distance around, can be written as twice the length plus twice the width, 38 = 2l+2w. In order to solve for width w, use the equation l = w + 5 to substitute into the perimeter equation:
38 = 2(w + 5) + 2w
38 = 2w + 10 + 2w
38 − 10 = 4w + 10 − 10
\(\frac{28}{4}=\frac{4w}{4}\)
w = 7

To convert 25°C to degrees Fahrenheit using the formula F=(⁹⁄₅)C+32,
simply substitute 25 for C in the formula:
(⁹⁄₅)25 + 32 = 77°F

As k gets larger and larger without bound, the expression \(\frac{4}{(2 + k)}\) becomes 4 divided by an increasingly large number. For example, think about the trend between the following fractions:\(\frac{4}{100},\frac{4}{10,000},\frac{4}{1,000,000}\),... Looking at it this way, you can see that the expression for r gets closer and closer to zero.

To solve this problem it is necessary to factor the polynomial x² − 2x − 24. In this case, since the final term is negative, you know that when this equation is factored, one sign will be positive and the other will be negative. Further, since the term is −24, the numbers in the problem will be factors of 24. The factors of 24 are 1 and 24, 2 and 12, 3 and 8, or 4 and 6. It is logical to look for the factors that have a difference of 2 because of the −2x term in the original polynomial. In this case, 4 and 6 are the only factors of 24 that have a difference of 2. Because the term −2x is negative, the logical expression is (x + 4)(x − 6). Therefore, x + 4 is the correct answer choice.

A square with sides (3x − 2) inches would have an area of (3x − 2)(3x − 2) square inches. Use the “FOIL” method to multiply all terms in the equation:
First: (3x)(3x) = 9x²
Outside: (3x)(−2) = −6x
Inside: (−2)(3x) = −6x
Last: (−2)(−2) = 4
Adding all of these terms yields the expression 9x² − 6x − 6x + 4. After combining like terms, the expression is 9x² − 12x + 4.

C. The line 3x + 4y = −12 is given in standard form. To find the slope, convert the equation into slope-intercept form (y = mx+b, where m is the slope and b is the y-intercept):
(3x + 4y) − 3x = −12 − 3x
\(\frac{(4y)}{4}=\frac{(−3x − 12)}{4}\)
y =(^-3⁄₄)x − 3
Since the equation is now in slope-intercept form, it is easy to see that the slope is ^-3⁄₄.