SOLUTION: Solve. 4^x=(1/square root of 2)

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Question 598192: Solve.
4^x=(1/square root of 2)
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--
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4%5Ex=1%2Fsqrt%282%29
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Multiply the right hand side of the equation by sqrt%282%29%2Fsqrt%282%29 to clear the radical expression in the denominator.
4%5Ex=%281%2Fsqrt%282%29%29%2A%28sqrt%282%29%2Fsqrt%282%29%29
4%5Ex=sqrt%282%29%2F2
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Rewrite each term in the equation in terms of base 2 using exponent rules. 4=2^2, sqrt(2)=2^(1/2), and 2=2^1.
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4%5Ex=sqrt%282%29%2F2
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When you raise an exponential expression to a power, you multiply the exponents, so (2^2)^x=2^2x.
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When you divide exponential expressions, you subtract the exponents, so 2^(1/2)/2^1=2^(1/2-1)=2^(-1/2)
%282%5E2%29%5Ex=2%5E%281%2F2%29%2F2%5E1%29
2%5E%282x%29=2%5E%28-1%2F2%29
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We have 2 raised to a power on both sides of the equation. Since both expressions are equal, the exponents must be the same, So
2x=-1%2F2
x=-1%2F4
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You definitely want to check your work on these sorts of problems. Use the original equation.
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4%5Ex=1%2Fsqrt%282%29
4%5E%28-1%2F4%29=1%2Fsqrt%282%29
1%2F4%5E%281%2F4%29=1%2Fsqrt%282%29
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4^(1/4) means the fourth root of 4. The fourth root of 4 is the square root of 2 since (sqrt(2))(sqrt(2))(sqrt(2))(sqrt(2))=4.
1%2Fsqrt%282%29=1%2Fsqrt%282%29
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That's it. x is -1/4.
You can also solve this using logarithms. Feel free to email if you'd like to see that solution.
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Ms.Figgy
math.in.the.vortex@gmail.com