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Simplify. Assume xx is greater than or equal to zero.\newline45x9\sqrt{45x^9}

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Full solution

Q. Simplify. Assume xx is greater than or equal to zero.\newline45x9\sqrt{45x^9}
  1. Factorization: Factor 45x945x^9 into its prime factors and perfect squares.\newline4545 can be factored into 9×59 \times 5, where 99 is a perfect square. The x9x^9 term can be written as (x4)2×x(x^4)^2 \times x, where (x4)2(x^4)^2 is a perfect square.\newlineSo, 45x9=9×5×(x4)2×x45x^9 = 9 \times 5 \times (x^4)^2 \times x.
  2. Rewrite Expression: Rewrite the square root of the expression using the factorization. 45x9=9×5×(x4)2×x\sqrt{45x^9} = \sqrt{9 \times 5 \times (x^4)^2 \times x}.
  3. Simplify Square Root: Simplify the square root by taking out the perfect squares. 9×5×(x4)2×x=9×(x4)2×5x\sqrt{9 \times 5 \times (x^4)^2 \times x} = \sqrt{9} \times \sqrt{(x^4)^2} \times \sqrt{5x}. This simplifies to 3×x4×5x3 \times x^4 \times \sqrt{5x}.
  4. Combine Terms: Combine the simplified terms outside of the square root. The final simplified expression is 3x4⋅5x3x^4 \cdot \sqrt{5x}.

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