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both positive, write the followi

y^(3)sqrt(45x^(7)y^(5))

both positive, write the followi\newliney345x7y5 y^{3} \sqrt{45 x^{7} y^{5}}

Full solution

Q. both positive, write the followi\newliney345x7y5 y^{3} \sqrt{45 x^{7} y^{5}}
  1. Break down square root: First, let's break down the square root of 4545 into prime factors: 45=9×5=9×5=35\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}.
  2. Apply exponent properties: Now, let's apply the properties of exponents to the xx and yy terms inside the square root: x7y5=x72y52\sqrt{x^{7}y^{5}} = x^{\frac{7}{2}}y^{\frac{5}{2}}.
  3. Combine coefficients and terms: Combine the coefficients and the like terms: y3×35×x72y52y^{3} \times 3\sqrt{5} \times x^{\frac{7}{2}}y^{\frac{5}{2}}.
  4. Multiply y terms: Multiply the y terms together using the property of exponents ya×yb=ya+by^{a} \times y^{b} = y^{a+b}: y3×y52=y3+52=y112y^{3} \times y^{\frac{5}{2}} = y^{3 + \frac{5}{2}} = y^{\frac{11}{2}}.
  5. Multiply coefficients and combine: Now, multiply the coefficients (3×53 \times \sqrt{5}) and combine all terms: 35x72y1123\sqrt{5}x^{\frac{7}{2}}y^{\frac{11}{2}}.