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Question:

\(x^6 x^5 \) = `____`

A \(x^{11}\)

explanation

Recall that the multiplication of terms sharing the same base can be simplified by adding the exponents: \(x^a x^b = x^{a+b}\) .
The expression in this case, then, can be simplified as: \(x^5 x^6 = x^{a+b} = x^{11}\).

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Solve for x:
3(x + 1) = 5(x − 2) + 7.

3

Begin by distributing the 3 and the 5 through their respective parentheses, then combine like terms on each side of the equal sign:
3(x + 1) = 5(x − 2) + 7
3x + 3 = 5x − 10 + 7
3x + 3 = 5x − 3
3x + 6 = 5x
6 = 2x
3 = x

If \(y=3ab+2 b^3 \) , what is y when a = 1 and b = 2?

22

Substitute the given values for a and b into the equation and then evaluate to find y: \(y=3ab+2 b^3 \)
\(y=3 \times 1 \times 2+2 \times 2^3 \)
y = 6 + 2(8)
y = 6 + 16
y = 22.

Calculate the area of the circle shown in the figure.

100π cm^2.

The area of a circle is equal to \( \pi R^2 \) , and with a radius of 10 cm, the area is equal to 100 square centimeters multiplied by π.

Which of the following statements about trapezoids is true?

A trapezoid can have 4 angles which are not equal to one another.

Trapezoids do not need to have any angles that are equal to one another. The formula \( \frac{\mathrm{(a+b) \times h} }{\mathrm{2} }\) is for area of a trapezoid. Trapezoids can have right angles but do not have to have a 90 degree angle. This answer is correct because it is possible for a trapezoid to have 4 different angles.

Two parallel lines are intersected by a third line, which is perpendicular to the first parallel line. The value of another angle made by the intersecting lines is ____

90 degrees

If the first line is intersected and forms a 90 degree angle, that means that all other angles formed from the intersected line are also equal to 90 degrees.

A duck pond has 350 ducks. If 75 go south for the winter, how many ducks will still be in the pond?

275

To calculate this value, subtract the number of ducks that left (75) from the number of ducks that were originally there (350): 350 – 75 = 275

44 is 80% of what number?

55

Translate the question into an equation, using x to represent the unknown number:
44 = 0.8x
Divide both sides by the decimal:
\(\frac{\mathrm{44} }{\mathrm{0.8} }\) = x
x = 55.

A woman buys 23 pounds of potatoes. She gives 10% of them to a neighbor and 15% of them to her mother. How many pounds does she have left?

17.25 pounds

First, calculate 10% of 23 pounds: \(23 \times 10 \% = 2.3\) pounds
Then, calculate 15% of 23 pounds: \(23 \times 15 \% = 3.45\)
Then, calculate the total number of pounds the woman gave away: 2.3 + 3.45 = 5.75
Finally, calculate how many pounds she has left: 23 – 5.75 = 17.25

Given that x = 0.5 and y = 4, what is the value of \(4 x^2 (2y + 2)\)

10

The value of \(|x^2| \) is x × x = 0.5 × 0.5 = 0.25, and as a result \(|4x^2|\) is equal to 1. This means the equation simplifies to 1(2y + 2). Since y has a value of 4, if we substitute and solve we find the answer is 10.

In the figure below, lines d and f are parallel and the angle measures are as given. What is the value of x ?

85

Vertical angles are the angles opposite of each other that are created when two lines cross. Vertical angles are always equal. So, the angle vertical to the 35-degree angle also has a measure of 35 degrees. The supplement of the 120-degree angle has a measure of 60 degrees (all straight lines measure 180 degrees), so we then have a triangle with angles measuring 35, 60, and x degrees, as shown in the figure provided. Since the angles of a triangle add to 180 degrees, x = 180 − 35 – 60 = 85 degrees.