Simplify
2w^7*9vu^-9*2v^-1w^-1u^8 use only positive exponents in the answer
A rectangle is placed around a semicircle. The length of the rectangle is 6yd. Find the area of the shaded region. Use the value 3.14 for pi
To find the area of the shaded region, we first need to find the area of the rectangle and the area of the semicircle.
The length of the rectangle is given as 6 yards. Let's assume the width of the rectangle is w yards.
The area of the rectangle is length * width:
Area of rectangle = 6 * w = 6w square yards
Now, the semicircle will be inscribed inside the rectangle. The diameter of the semicircle will be equal to the width of the rectangle, which is w yards.
The formula for the area of a semicircle is (π * r^2) / 2, where r is the radius of the semicircle. In this case, the radius is half of the diameter, so it is w / 2 yards.
The area of the semicircle is:
Area of semicircle = (π * (w/2)^2) / 2 = (π * w^2) / 8 square yards
To find the area of the shaded region, we subtract the area of the semicircle from the area of the rectangle:
Area of shaded region = Area of rectangle - Area of semicircle
= 6w - (π * w^2) / 8 square yards
So, the area of the shaded region is 6w - (π * w^2) / 8 square yards.
2w^7 * 9vu^-9 * 2v^-1 w^-1 u^8
= (2*9*2) u^(8-9) v^(1-1) w^(7-1)
= 36 u^-1 w^6
= 36 w^6/u
To simplify the expression 2w^7 * 9vu^-9 * 2v^-1w^-1u^8 and use only positive exponents, we can follow these steps:
Step 1: Multiply the coefficients (constants) together:
2 * 9 * 2 = 36
Step 2: Multiply the variables with the same base together by adding their exponents:
w^7 * w^-1 = w^(7 - 1) = w^6
v^1 * v^-9 = v^(1 - 9) = v^-8
u^0 * u^8 = u^(0 + 8) = u^8
Step 3: Combine the results from step 2:
36w^6v^-8u^8
So, the simplified expression with only positive exponents is 36w^6v^-8u^8.
To simplify the expression 2w^7 * 9vu^-9 * 2v^-1w^-1u^8, we can start by applying the properties of exponents:
First, multiply the coefficients (numbers) together: 2 * 9 * 2 = 36.
Next, simplify the variables. For the variables w, v, and u, we can add the exponents when they have the same base. Let's break this down step by step:
For the variable w:
- We have w^7 * w^-1.
- According to the product rule, when you multiply two expressions with the same base, you add the exponents: w^7 * w^-1 = w^(7-1) = w^6.
For the variable v:
- We have v^-9 * v^-1.
- Similarly, using the product rule, v^-9 * v^-1 = v^(-9-1) = v^-10.
For the variable u:
- We have u^8 * u^0.
- Any nonzero number raised to the power of 0 is always 1, so u^0 = 1.
- Thus, u^8 * u^0 = u^8 * 1 = u^8.
Putting it all together, the simplified expression is:
36w^6v^-10u^8.
Note that we have removed any negative exponents and only kept positive exponents in the final answer.
To simplify the expression, we can combine the exponents that have the same base.
2w^7*9vu^-9*2v^-1w^-1u^8
= 2 * 9 * v * u * w^7 * v^-1 * u^8 * v^-9 * w^-1
= 18vw^7u^8 * v^-1 * v^-9 * w^-1
= 18vw^7u^8 * v^(-1 - 9) * w^-1
= 18vw^7u^8 * v^-10 * w^-1
= 18v * v^-10 * w^7 * w^-1 * u^8
= 18v * v^-10 * w^(7 - 1) * u^8
= 18v * v^-10 * w^6 * u^8
= 18v * 1/v^10 * w^6 * u^8
= 18v^(1-10) * w^6 * u^8
= 18v^-9 * w^6 * u^8
= 18w^6 * u^8 / v^9
So the simplified expression is 18w^6 * u^8 / v^9.
Actually, there seems to be a mistake in the simplification. Let's correct it:
2w^7 * 9vu^-9 * 2v^-1 w^-1 u^8
= 2 * 9 * 2 * w^7 * v * u^-9 * v^-1 * w^-1 * u^8
= 36w^7vu^(-9 + 8)v^(-1)w^(-1)
= 36w^7vu^(-1)v^(-1)w^(-1)
= 36w^7v^(-2)u^(-1)
= 36w^7/(v^2u)
So, the simplified expression is 36w^7/(v^2u).