How do you solve this?
(x-2)^(5/2)=243
Can someone explain this to me?
(x-2)^(5/2)=243
Can someone explain this to me?
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243 = 3^5 so,
(x-2)^(5/2)=243 [Raise each side to the power of 2/5] iff
[(x - 2)^(5/2)]^(2/5) = (3^5)^(2/5) [Power of power: (a^m)^n = a^(mn)] iff
(x - 2)^(5/2 * 2/5) = 3^(5 * 2/5) iff
(x - 2)^1 = 3^2 iff
x - 2 = 9 iff
x = 11
(x-2)^(5/2)=243 [Raise each side to the power of 2/5] iff
[(x - 2)^(5/2)]^(2/5) = (3^5)^(2/5) [Power of power: (a^m)^n = a^(mn)] iff
(x - 2)^(5/2 * 2/5) = 3^(5 * 2/5) iff
(x - 2)^1 = 3^2 iff
x - 2 = 9 iff
x = 11
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This is an order of operations problem, so (x-2)^(5/2) is all "one number" with multiple pieces, the power of (5/2) is the last piece of the equation, so it must be take out first.
this can be done by taking both sides to the (2/5) power
((x-2)^((5/2)(2/5)))= (243^(2/5))
so now
((x-2)^1)=9
anything to the first power is itself so
x-2=9
therefor
x=11
sorry about using so many parenthesis, but I'm used to putting equations into computer programs and parenthesis are a way of defining the order of operation. Hope this helps.
this can be done by taking both sides to the (2/5) power
((x-2)^((5/2)(2/5)))= (243^(2/5))
so now
((x-2)^1)=9
anything to the first power is itself so
x-2=9
therefor
x=11
sorry about using so many parenthesis, but I'm used to putting equations into computer programs and parenthesis are a way of defining the order of operation. Hope this helps.
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like any algebra problem you need to find the value of x, but you know that already ;)
i'm gonna guess that what's confusing you is the exponent (power) that's a fraction.
to understand what having a fraction as an exponent means, we need to combine something you already know with something you don't.
this is the part you probably don't know: x^(1/n) means you take the nth root of x. for example, if we had 4^(1/2), the 2 on the bottom of the fraction means we do a square root. that would be the same as the square root of 4, which is 2. 8^(1/3) has a 3 on the bottom, so that would be the cube root of 8, which is also 2. (2^3 = 8)
so, the bottom number of the fraction means a root. but what about the top number? what if the number on the top of the fraction isn't 1? well, the number on the top of the fraction acts the same as a regular exponent. so the 2 in x^(2/3) and x^2 does the same thing, it squares x.
let's pretend we want to find out what 4^(2/4) equals. the 2 means we square 4 to get 16, while the /4 means we take the 4th root of 16, which is 2. (the order doesn't matter, you can do the root first and the power second and you will get the same answer.)
back to your question: it's saying that (x-2) to the power of (5/2) equals 243. don't worry about the -2 part until the end. find out what number to the power (5/2) makes 243, then you can worry about the x-2 part.
i'm gonna guess that what's confusing you is the exponent (power) that's a fraction.
to understand what having a fraction as an exponent means, we need to combine something you already know with something you don't.
this is the part you probably don't know: x^(1/n) means you take the nth root of x. for example, if we had 4^(1/2), the 2 on the bottom of the fraction means we do a square root. that would be the same as the square root of 4, which is 2. 8^(1/3) has a 3 on the bottom, so that would be the cube root of 8, which is also 2. (2^3 = 8)
so, the bottom number of the fraction means a root. but what about the top number? what if the number on the top of the fraction isn't 1? well, the number on the top of the fraction acts the same as a regular exponent. so the 2 in x^(2/3) and x^2 does the same thing, it squares x.
let's pretend we want to find out what 4^(2/4) equals. the 2 means we square 4 to get 16, while the /4 means we take the 4th root of 16, which is 2. (the order doesn't matter, you can do the root first and the power second and you will get the same answer.)
back to your question: it's saying that (x-2) to the power of (5/2) equals 243. don't worry about the -2 part until the end. find out what number to the power (5/2) makes 243, then you can worry about the x-2 part.
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(x-2)^5 = 243^2 = 59049
x-2 = 59049^(1/5) = 9
x = 11
x-2 = 59049^(1/5) = 9
x = 11