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sqrt(rs)sqrt(5r^(2)s)sqrt(5rs)-sqrt(15r^(3)s)sqrtrsqrt(135s^(2))

rs5r2s5rs−15r3sr135s2 \sqrt{r s} \sqrt{5 r^{2} s} \sqrt{5 r s}-\sqrt{15 r^{3} s} \sqrt{r} \sqrt{135 s^{2}}

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Q. rs5r2s5rs−15r3sr135s2 \sqrt{r s} \sqrt{5 r^{2} s} \sqrt{5 r s}-\sqrt{15 r^{3} s} \sqrt{r} \sqrt{135 s^{2}}
  1. Combine Square Roots: First, let's simplify each square root separately by combining them where possible and simplifying any perfect squares.\newlineWe start with the first term rs\sqrt{rs}5r2s\sqrt{5r^2s}5rs\sqrt{5rs}.
  2. Multiply Inside Square Root: Combine the square roots in the first term using the property ab=ab\sqrt{a}\sqrt{b} = \sqrt{ab}:rs×5r2s×5rs=rs×5r2s×5rs\sqrt{rs} \times \sqrt{5r^2s} \times \sqrt{5rs} = \sqrt{rs \times 5r^2s \times 5rs}.
  3. Simplify Inside Square Root: Now, multiply the terms inside the square root: rs×5r2s×5rs=5×5×r×r2×s×s×r×s\sqrt{rs \times 5r^2s \times 5rs} = \sqrt{5 \times 5 \times r \times r^2 \times s \times s \times r \times s}.
  4. Evaluate Perfect Squares: Simplify the expression inside the square root: 25×r4×s3=25×r4×s3\sqrt{25 \times r^4 \times s^3} = \sqrt{25} \times \sqrt{r^4} \times \sqrt{s^3}.
  5. Simplify First Term: Evaluate the square roots of the perfect squares: 25=5\sqrt{25} = 5, r4=r2\sqrt{r^4} = r^2, and s3\sqrt{s^3} cannot be simplified further. So, the first term becomes 5×r2×s35 \times r^2 \times \sqrt{s^3}.
  6. Combine Square Roots: Now, let's simplify the second term 15r3s\sqrt{15r^3s}r\sqrt{r}135s2\sqrt{135s^2}. Combine the square roots using the property a\sqrt{a}b=ab\sqrt{b} = \sqrt{ab}: 15r3s×r×135s2=15r3s×r×135s2\sqrt{15r^3s} \times \sqrt{r} \times \sqrt{135s^2} = \sqrt{15r^3s \times r \times 135s^2}.
  7. Multiply Inside Square Root: Multiply the terms inside the square root: 15r3s×r×135s2=15×135×r3×r×s×s2.\sqrt{15r^3s \times r \times 135s^2} = \sqrt{15 \times 135 \times r^3 \times r \times s \times s^2}.
  8. Simplify Inside Square Root: Simplify the expression inside the square root: 2025×r4×s3=2025×r4×s3\sqrt{2025 \times r^4 \times s^3} = \sqrt{2025} \times \sqrt{r^4} \times \sqrt{s^3}.
  9. Evaluate Perfect Squares: Evaluate the square roots of the perfect squares: 2025=45\sqrt{2025} = 45, r4=r2\sqrt{r^4} = r^2, and s3\sqrt{s^3} cannot be simplified further. So, the second term becomes 45×r2×s345 \times r^2 \times \sqrt{s^3}.
  10. Simplify Second Term: Now we have the simplified forms of both terms:\newlineFirst term: 5×r2×s35 \times r^2 \times \sqrt{s^3}\newlineSecond term: 45×r2×s345 \times r^2 \times \sqrt{s^3}\newlineSubtract the second term from the first term:\newline5×r2×s3−45×r2×s35 \times r^2 \times \sqrt{s^3} - 45 \times r^2 \times \sqrt{s^3}.
  11. Subtract Terms: Factor out the common factor r2s3r^2 \sqrt{s^3}:\newliner2s3×(5−45)r^2 \sqrt{s^3} \times (5 - 45).
  12. Factor Out Common Factor: Subtract the numbers inside the parentheses:\newline5−45=−405 - 45 = -40.\newlineSo, the expression simplifies to:\newline−40×r2×s3-40 \times r^2 \times \sqrt{s^3}.

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