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hello, i know it is to late but u can easily solve it easier with the used of substitution you can let u = 250 then it will result to ((2u)^(2u))/(u^u) (this looks better if you write it on paper) and simplifying it would result to (4u)^u and substituting it back will be 1000^250
I really like your videos. Easy pace and very well narrated for everyone to follow. The writing is very neat and clear. I would love to know what pen and nib size you use!!
Alternate method : 500^500=(5x10^2)^500 --->>5^500 x 10^1000 250^250=(5^2x10)^250--->>5^500 x 10^250 Put these last values in the question:--- *). 5^500 will get canceled with the 5^500 in the denominater and *). In tyhe end you will be left with ------>>>> 10^1000/10^250 Which will become 10^(1000-250) == 10^750
Got bored and decided to make a general rule for (b^b)/(a^a), where a&b are whole numbers and b>a: [b^(b-a)]*[(b/a)^a] Proof: [b^(b-a+a)] / [a^a] [b^(b-a)]*[b^a] / [a^a] [b^(b-a)] * [b^a]/[a^a] [b^(b-a)]*[(b/a)^a] Example using b = 3, and a = 2: [b^(b-a)]*[(b/a)^a] [3^(3-2)]*[(3/2)^2] [3^(1)]*[(9/4)] [3]*[9/4] 27/4 In this case b = 2*a so you could simplify further. [(2a)^(2a-a)]*[(2a/a)^a] [(2a)^(a)]*[(2)^a] (4a)^a (4*250)^250 1000^250 = 10^750
We can choose another way. 500^(500)= 250^500 .2^500. But 250^500=250^250 .250^250. The next one is 250^250 .2^500. But 250^250= 2^250 . 125^250. But 125^250= (5^3)^250= 5^750. 2^500.2^250=2^750. 2^750.5^750= 10^750
I got a different answer. The value of 500 ^ 500 / 250 ^ 250 is 2 ^ 500, which is approximately..... 1.6069380442589904 × 10^150. The proof is as follows. We can simplify this expression by using exponent laws. First, we can write 500 as 2 × 250, so we have: 500 ^ 500 = (2 × 250) ^ 500 We can use the power of a product rule to expand this expression: (2 × 250) ^ 500 = 2 ^ 500 × 250 ^ 500 Similarly, we can write 250 as 2 × 125, so we have: 250 ^ 250 = (2 × 125) ^ 250 Using the power of a product rule again, we get: (2 × 125) ^ 250 = 2 ^ 250 × 125 ^ 250 Now we can substitute these expressions back into the original expression and simplify: 500 ^ 500 / 250 ^ 250 = (2 ^ 500 × 250 ^ 500) / (2 ^ 250 × 125 ^ 250) We can cancel out the common factor of 2 ^ 250 in the numerator and denominator: 500 ^ 500 / 250 ^ 250 = 2 ^ 250 × (250 / 125) ^ 250 Simplifying 250 / 125 to 2, we get: 500 ^ 500 / 250 ^ 250 = 2 ^ 250 × 2 ^ 250 = 2 ^ 500 Therefore, the value of 500 ^ 500 / 250 ^ 250 is 2 ^ 500, which is approximately 1.6069380442589904 × 10^150.
