1.Simplify the expression below
(-2w^3)^5 / 8w. What is the value of the exponent on "w" ?
2.Simplify the expression (4x^2 - 3x +1) - (x^3 +2x +7). What is the coefficient of the x term in the simplfied expression?
3. Jack Watson is wroking in a lab making microchips for computers. he uses a "super tool" that measures the size of a transitor on a microchip by magnifying the transitor to twice its actual size and measuring the magnified size. The actual size of the transitor must be within 4 nanometers of the magnified industry standard. Theindustry standard for the actual size is 80 nanometers. The inequality below models the range of the actual size of the transitor.
-4 <or equal to 2(x -80)<4. Which graph correctly represents the range in which the actual transitor size for a particular transitor is within the industry standard ?
4.Use line with equations x + 5y = 5 and 5x + py = 5.
a. Find p if the lines are parallel
b. Find p if the lines are perpendicular
5.A line passes through the points (6,4) and (5,3). What is the equation of the line in point-slope form ?
a. x-6 = 7(y-4)
b. y-4 = 7(x-6)
c. y+6 = 7(x+4)
d. y-6 = 7(x-4)
Can you please solve these problems clearly so that I can follow it to understand, thanks, Please help me.
I will be happy to check your answers.
To rbowh
I do not know how to solve them, that is why I'm asksing for help,
1. To simplify the expression (-2w^3)^5 / 8w, we need to simplify each component separately. Let's start with (-2w^3)^5:
The exponent outside the parentheses applies to everything inside. Therefore, we raise -2w^3 to the power of 5:
(-2w^3)^5 = -2^5 * (w^3)^5 = -32 * w^(3*5) = -32w^15
Next, we divide (-32w^15) by 8w:
(-32w^15) / 8w = -32/8 * w^15/w = -4w^14
So, the simplified expression is -4w^14. The value of the exponent on "w" is 14.
2. To simplify the expression (4x^2 - 3x + 1) - (x^3 + 2x + 7), we combine like terms:
Group the terms with the same exponent:
4x^2 - x^3 - 3x + 2x + 1 - 7
Now, rearrange the terms:
- x^3 + 4x^2 - 3x + 2x + 1 - 7
Combine the x^3 and x^2 terms:
- x^3 + 4x^2 - x + 1 - 7
Combine the x and constant terms:
- x^3 + 4x^2 - x - 6
The coefficient of the x term in the simplified expression is -1.
3. To determine the correct graph representing the range in which the actual transistor size is within the industry standard, we need to solve the inequality:
-4 ≤ 2(x - 80) ≤ 4
Start by simplifying the inequality:
-4 ≤ 2x - 160 ≤ 4
Add 160 to all parts of the inequality:
160 - 4 ≤ 2x - 160 + 160 ≤ 4 + 160
156 ≤ 2x ≤ 164
Divide all parts of the inequality by 2:
78 ≤ x ≤ 82
Therefore, the actual size of the transistor must be within the range of 78 to 82 nanometers. The correct graph would show a shaded region between these two values on the number line.
4. For the line equations x + 5y = 5 and 5x + py = 5:
a. To find p when the lines are parallel, we need to compare the slopes. Both equations must be in slope-intercept form (y = mx + b) to easily determine their slopes.
From the equation x + 5y = 5, rearrange it to y = -(1/5)x + 1. The slope of this line is -1/5.
From the equation 5x + py = 5, rearrange it to y = -(5/p)x + 1. To make the lines parallel, the slopes must be the same, so -(5/p) must also be -1/5. Solve for p:
-(5/p) = -1/5
Cross-multiply:
-5 = -p
Therefore, p = 5.
b. To find p when the lines are perpendicular, we need to compare the slopes again.
The slope of the first equation, x + 5y = 5, is -1/5.
To make the lines perpendicular, the slopes must be negative reciprocals of each other. Therefore, the slope of the second equation, 5x + py = 5, must be 5.
Now, by rearranging 5x + py = 5 to y = -(5/p)x + 1, we can compare the slopes:
-(5/p) = 5
Solve for p:
5/p = -5
Cross-multiply:
-25 = p
Therefore, p = -25.
5. To find the equation of the line passing through the points (6,4) and (5,3) in point-slope form, we need to use the formula:
y - y1 = m(x - x1), where (x1, y1) represents one of the points and m is the slope of the line.
First, find the slope (m):
m = (y2 - y1) / (x2 - x1)
Using the points (6,4) and (5,3), we have:
m = (3 - 4) / (5 - 6) = -1 / -1 = 1
Now, substitute the slope (m) and one of the points (x1, y1) into the equation:
Using (6,4):
y - 4 = 1(x - 6)
Rearrange the equation to point-slope form:
y - 4 = x - 6
Simplify:
y = x - 2
Therefore, the equation of the line passing through the points (6,4) and (5,3) is y = x - 2.