\nThe problem statement is “$\\sqrt{x}$% of $x$ is 80”. We need to translate this sentence into a mathematical equation.
\n“$\\sqrt{x}$%” means $\\frac{\\sqrt{x}}{100}$.
\n“of $x$” means multiply by $x$.
\n“is 80” means equals 80.
\nSo the equation is:
\n$\\frac{\\sqrt{x}}{100} \\times x = 80$.
\nRewrite $\\sqrt{x}$ as $x^{1\/2}$ and $x$ as $x^1$:
\n$\\frac{x^{1\/2} \\times x^1}{100} = 80$.
\nUsing the rule of exponents $a^m \\times a^n = a^{m+n}$, combine the terms with x:
\n$\\frac{x^{1\/2 + 1}}{100} = 80$
\n$\\frac{x^{3\/2}}{100} = 80$.
\nMultiply both sides by 100:
\n$x^{3\/2} = 80 \\times 100 = 8000$.
\nTo solve for x, raise both sides of the equation to the power of (2\/3), which is the reciprocal of 3\/2:
\n$(x^{3\/2})^{2\/3} = (8000)^{2\/3}$.
\n$x^1 = (8000)^{2\/3}$.
\n$(8000)^{2\/3}$ can be calculated as $(8000^{1\/3})^2$ or $(8000^2)^{1\/3}$. It’s usually easier to find the cube root first.
\nWe need to find the cube root of 8000. $8000 = 8 \\times 1000 = 2^3 \\times 10^3 = (2 \\times 10)^3 = 20^3$.
\nSo, $8000^{1\/3} = 20$.
\nNow, square the result:
\n$x = (20)^2 = 400$.
\nLet’s verify the answer: $\\sqrt{400} = 20$. 20% of 400 = $\\frac{20}{100} \\times 400 = 0.20 \\times 400 = 80$. The result matches the given condition.
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