```Question 604741
In the work that follows the software that renders equations to look like you would write them. The software has some troubles with handling fractional exponents and radical signs in exponents. In the case of radical signs, they appear to look more like just a check mark because they are lacking the horizontal line above the quantity that is within the radical. You'll have to make a corrective adjustment in picturing the way it should be. Sorry about that. In other situations fractional exponents get clipped off. I'll note where this happens so that you can understand what the exponent should look like. Again, my apologies.

The book answer shows that two values of x will make the left side of this equation equal to the right side:
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{{{9^sqrt(x)= 3^x}}} <--- note that the top of the radical sign is likely missing
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The first solution (x = 0)is intuitively obvious. It involves the fact that if any number is raised to the exponent zero, the answer will be 1. Note that both exponents in this equation will be zero if x is set equal to zero. With x equal to zero the equation becomes:
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{{{9^sqrt(0)= 3^0}}} <---- again, the top bar of the radical sign is likely gone
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Since the square root of zero is zero, the left side of this equation becomes as shown below:
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{{{9^0 = 3^0}}}
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and with both exponents now 0 the equation becomes:
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{{{1 = 1}}}
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This obviously is true, so when x equals zero, the equation is true. Therefore, x = 0 is one solution to the problem.
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Finding the other solution is a little less obvious. Start with the given equation:
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{{{9^sqrt(x) = 3^x}}} <---- again the top bar of the radical is likely missing
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Now instead of writing the exponent of 9 with a radical sign, let's change the exponent so that x has an exponent of 1/2 itself. An exponent of 1/2 is equivalent to the square root radical. That being the case we can write the equation in the equivalent form:
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{{{(9)^x^(1/2) = 3^x}}} <---- the 1 in the numerator of the exponent 1/2 is likely clipped off a little
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But we know that 9 is equal to 3 squared (or {{{9 = 3^2}}}). Substitute this for 9 and the equation becomes:
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{{{(3^2)^x^(1/2) = 3^x}}}
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Now using the power rule of exponents, we can multiply the two exponents (2 and x^(1/2)) on the left side to make the equation become:
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{{{(3)^(2*x^(1/2))= 3^x}}} <---- on the left side the x in the exponent has an exponent of 1/2. It is clipped so that only some of the 2 is visible. Think of it as 1/2
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Note now that the bases on both sides of this equation are 3. Therefore, for the sides to be equal, the exponents have to be equal. In other words we can set the exponents on both sides equal to each other to get the equation:
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{{{2x^(1/2) = x}}} <--- the numerator 1 is likely to be slightly clipped off
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Square both sides of this equation and you get:
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{{{4x = x^2}}}
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Divide both sides by x and the equation then becomes:
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{{{4 = x}}}
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This tells you that both sides of the original equation become equal if x equals 4. You can check this by returning to the original equation:
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{{{9^sqrt(x) = 3^x}}} <---- the radical sign is missing the top horizontal line
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Substitute 4 for x and you get:
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{{{9^sqrt(4) = 3^4}}} <---- and again the radical sign is missing its top
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But the square root of 4 is 2 so the equation becomes:
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{{{9^2 = 3^4}}}
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Square 9 and you get 81 on the left side. Then if you raise 3 to the fourth power you also get 81 on the right side. This means that the answer x = 4 satisfies the equation by making both sides equal and therefore x = 4 is a valid solution.
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Hope this helps you to see how to get the two answers to this problem.
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