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# helppppp

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1. If $$5^x=100$$, what is the value of $$5^{x+2}$$?

2. A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?

Aug 21, 2018

#2
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1. No need to find the value of x. Using the laws of indices:

$$5^{x+2}=5^x\times5^2\rightarrow 5^x\times25\rightarrow100\times25\rightarrow2500$$

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Aug 21, 2018

#1
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1. If $$5^x=100$$ , what is the value of $$5^{x+2}$$?

$$5^x= 100\\ x\cdot ln5 = ln100\\ x= \color{blue}\frac{ln100}{ln5}$$

$$5^{x+2}=5^{\frac{ln100}{ln5}+2}$$

$$5^{x+2}=2500$$

!

Aug 21, 2018
edited by asinus  Aug 21, 2018
#2
+27908
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1. No need to find the value of x. Using the laws of indices:

$$5^{x+2}=5^x\times5^2\rightarrow 5^x\times25\rightarrow100\times25\rightarrow2500$$

Alan Aug 21, 2018
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Thanks! Can you solve problem 2/

Aug 21, 2018
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2. Pythagorean triples are the right angled triangles with integer leg lengths:

Oops! I missed one (probably the most obvious one!!).

n = 4, m = 5     4(5 - 4) = 4           2mn = 40     m2-n2 =  9     m2+n2 = 41   Area = 180

Hence sum of all possible areas:  216 + 180 = 396

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Aug 21, 2018
edited by Alan  Aug 21, 2018