Question 465887
Given:
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{{{3+sqrt(z-8)= sqrt(z+7)}}}
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The next step is to square both sides. Squaring the left side is done by multiplying:
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{{{(3+sqrt(z-8))*(3+sqrt(z-8))}}}
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This can be done by using the FOIL method ... multiply First Terms, then Outside Terms, then Inside Terms, and finally Last Terms.  When you do that you get:
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{{{(3*3)+3*sqrt(z-8) +3*sqrt(z-8)+sqrt(z-8)*sqrt(z-8)}}}
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This simplifies to:
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{{{9 + 6*sqrt(z-8)+(z-8)}}} which further simplifies to:
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{{{9 + 6*sqrt(z-8)+z -8}}} 
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and the +9 and -8 combines to +1 so the expression reduces to:
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{{{6*sqrt(z-8) +z + 1}}}
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Now return to square the right side of the equation as follows:
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{{{sqrt(z+7)*sqrt(z+7) = z+7}}}
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So after squaring both sides the equation becomes:
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{{{6*sqrt(z-8) + z + 1 = z + 7}}}
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Subtract z from both sides reduces this to:
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{{{6*sqrt(z-8) + 1 = 7}}}
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Next subtract 1 from both sides to get:
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{{{6*sqrt(z-8) = 6}}}
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Divide both sides by 6:
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{{{sqrt(z-8) = 1}}}
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Square both sides:
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{{{z-8 = 1}}}
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Finally add 8 to both sides and the result is:
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{{{z = 9}}}
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That's the answer. You can check this by returning to the original equation, substituting 9 for z, and working it out to see that both sides are equal.  In other words, start with:
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{{{3+sqrt(z-8)= sqrt(z+7)}}}
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Substitute 9 for z:
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{{{3 + sqrt(9-8)=sqrt(9+7)}}}
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Combine the numbers under each radical to get:
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{{{3 + sqrt(1)=sqrt(16)}}}
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This reduces to:
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{{{3 + 1 = 4}}}
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And this is true (4 does equal 4) so our answer of z = 9 does satisfy the original equation.
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Hope this helps you to understand that squaring radicals is a way to solve problems such as these.