HAMBONEJOE

The exponent applies to the numerator and denominator separately. (1/2)² = 1²/2² = 1/4. For negative exponents, the numerator becomes the denominator and vice versa. (1/2)^-2 = 2²/1² = 4/1 = 4. Anything raised to the power of 0 is one. For fractional exponents, take the denominator root and raise that to the numerator. (1/2)^1/2 = square root of 1/2 raised to the power of 1. (1/2)^4/7 = seventh root of 1/2 raised to the fourth.

Multiply the bases and add the exponents. It's 15*10^13

|-3| = 3

7-3 = 4

If you're 2 and your sister is half your age, there is one year difference between you (You are older by 1 year). If you're 100, your sister is 100-1 or 99.

Divide both sides by 1500000

(1-c)^8 = 1/15

Take the eighth root

1-c = sqrt(1/15, 8)

Subtract 1

-c = sqrt(1/15, 8) - 1

Multiply by -1

c = 1 - sqrt(1/15, 8)

184 loaves in four hours/4 hours = 46 loaves in one hour

Look for factors of 360 that are perfect squares or use a prime factorization method. I'm terrible at finding perfect squares as factors so I normally just divide by prime numbers until I get a prime number, it's extremely inefficient but it's simple and always works. 360/2 is 180, 180/2 is 90, 90/2 is 45, 45/5 = 9, 9/3 is 3. We found 2*2*2*5*3*3 = 360.  So sqrt(360) is  sqrt(2) * sqrt(2) * sqrt(2) * sqrt(5) * sqrt(3) * sqrt(3). Sqrt(2) * sqrt(2) is 2, so every pair of numbers comes outside the radical (square root symbol), so 2*3 is 6 and we have sqrt(2) * sqrt(5) left, so the simplified version is 6*sqrt(10). The fastest way was to realize that 360 = 36*10, and the sqrt(36) is 6. An easy way to check the solution is to square the outside term and multiply it by the term inside the radical (6*6*10 = 360, so our answer is right).

What exactly are you asking?

Do I have to use the ln fact? u(x)/v(x) is like saying u(x) times v(x)^-1. For simplicity, I'll use u for u(x) and v for v(x) Now use the product rule:

d/dx [u * v^-1]

d/dx [u] * v^-1 + u * d/dx [v^-1] Use chain rule

u' * v​^-1 + u * (-1)(v)^-2 * v' 

u'/v - uv'/v²  Multiply u'/v by v to combine into a single denominator and:

(u'v - uv')/v² 

So:

d/dx [u/v] = (u'v - uv')/v² 

I could be wrong, but I think the answer is 165. Let's start with the 1000's first. For any number that starts with one and ends in two, the center two numbers must add to 2-1. Since the two center numbers have to add to 1 and the number starts with 1 and ends in 2, we get 1102 and 1012. It's similar to a number that starts with 1 and ends in a 3. The center two numbers must add to 3-1. There are 3 different ways 2 numbers can add to 2: 0,2 1,1 and 2,0. Therefore, we now have 1023, 1113, and 1203. Since there is 1 way to make a number that ends in 1, 2 ways that end in 2o, 3 ways for 3 etc, we can continue this pattern for 4 through 9, making a total of 1+2+3+4+5...+9, 45. So there are 45 four digit numbers that start with 1. For numbers that start with 2, we'll start with a number that starts with 2 and ends in 1, but 1-2 = -1, and we can't have two positive numbers that add to a negative number, so no number that starts with 2 and ends in 1 works. For numbers that start with 2 and end in 2, only 2002 works. For numbers that start with 2 and end in 3, we get two possible combinations, 0,1 and 1,0. Notice how as we increase our starting number the possibilities for each end number decrease by one from what the previous starting number is. Numbers that start with 1 had 1 combination for 1, 2 for 2,3 for 3, etc. For numbers that start with 2, we get 1 possibility for 2, 2 for 3, 3 for 4, etc. Since there are 9 numbers we can end with for starting with 1, there will be 8 ending numbers for 2, 7 for 3, etc. Since we loose 1 possibility each time we increase the starting number by 1, there will be 9 fewer combinations for numbers that start with 2, or 35. There will be 8 fewer numbers that start with 3 than 2, 7 fewer that start with 4 than start with 3 etc. So for all numbers 1000-9999, we started with 45 for 1, 45-9 for 2, 45-9-8 for 3, 45-9-8-7 for 4 and so on. 45+36+28+21+15+10+6+3+1 = 165. So (I think) there are 165 numbers from 1000-9999 that follow your rule.