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Simplify sqrt((1728)/(2025)) and rationalize the denominator.

Mathematics

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Answer

8315\frac{8\sqrt{3}}{15}

Step-by-step solution

5 steps

here's the solution for number 19.

19)

  1. Express 1728 in terms of its prime factors.

    1728=2×8641728 = 2 \times 864 =22×432= 2^2 \times 432 =23×216= 2^3 \times 216 =24×108= 2^4 \times 108 =25×54= 2^5 \times 54 =26×27= 2^6 \times 27 =26×33= 2^6 \times 3^3

  2. Express 2025 in terms of its prime factors.

    2025=5×4052025 = 5 \times 405 =52×81= 5^2 \times 81 =52×34= 5^2 \times 3^4

  3. Substitute the prime factorizations into the expression 17282025\sqrt{\frac{1728}{2025}}.

    17282025=26×3334×52\sqrt{\frac{1728}{2025}} = \sqrt{\frac{2^6 \times 3^3}{3^4 \times 5^2}}

  4. Simplify the expression under the square root.

    26×3334×52=2634−3×52\sqrt{\frac{2^6 \times 3^3}{3^4 \times 5^2}} = \sqrt{\frac{2^6}{3^{4-3} \times 5^2}} =2631×52= \sqrt{\frac{2^6}{3^1 \times 5^2}} =263×52= \frac{\sqrt{2^6}}{\sqrt{3 \times 5^2}} =26/23×52/2= \frac{2^{6/2}}{\sqrt{3} \times 5^{2/2}} =2353= \frac{2^3}{5\sqrt{3}} =853= \frac{8}{5\sqrt{3}}

  5. Rationalize the denominator.

    853×33=835×3\frac{8}{5\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{5 \times 3} =8315= \frac{8\sqrt{3}}{15} 8315\frac{8\sqrt{3}}{15}
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