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The correct value of the simplified expressions found using the laws of indices gives;d. [tex]\sqrt[4]{4}[/tex]a. [tex]\sqrt{\frac{5}{6} }[/tex]d. 3a. [tex]x\cdot \sqrt{y}[/tex]b. x²·yb. 1a. 1/315·x²·y⁴e. 6·x³·y⁵c. [tex]\frac{5}{9}[/tex]What are the laws of indices?The laws of indices state rules that enable the simplification of the combination of expressions of numbers and variables that are given in index form or involve powers of other numbers or variables1. [tex]3^{\frac{1}{4} } = \sqrt[4]{4}[/tex]The radical sign is used for expressing fractional indices.The correct option is d. [tex]\sqrt[4]{4}[/tex]2. The radical is in its simplest form when it cannot be further simplified, which gives the radical in its simplest form as option a. [tex]\sqrt{\dfrac{5}{6} }[/tex]3. The index of a number is the number to which the variable or number is raisedThe index of x³ in [tex]\sqrt{x^3}[/tex] is d. 34. The given expression has y raised to the power of [tex]\frac{1}{2}[/tex], which gives;[tex]x\cdot y^\frac{1}{2} = x\cdot \sqrt{y}[/tex]The correct option is a. [tex]x\cdot \sqrt{y}[/tex]5. The common index of the expression within the parenthesis is [tex]\frac{1}{2}[/tex], which gives;[tex](x^4\cdot y^2)^\frac{1}{2} = (x^{4\times \frac{1}{2} }\cdot y^{2 \times \frac{1}{2} })=x^2\cdot y[/tex]The correct option is b. x²·y6. The value of all numbers raised the power of zero is 1, therefore;[tex](2^{1/3}\cdot 3^{1/4})^0\cdot (2^{1/5}\cdot 3)^0 = 1 \times 1=1[/tex]The correct option is b. 17. The given expression can be simplified as follows;[tex]\sqrt[5]{\dfrac{1}{243} } = \sqrt[5]{\dfrac{1}{3^5} } = \sqrt[5]{\left(\dfrac{1}{3} \right)^5} = \dfrac{1}{3}[/tex]The correct option is therefore; a. 1/38. The given expression [tex]\sqrt{225\cdot x^4\cdot y^8} = \sqrt{15^2\cdot x^4\cdot y^8} = 15\cdot x^2\cdot y^4[/tex]The correct option is [tex]15\cdot x^2\cdot y^4[/tex]9. The given expression is [tex]\sqrt[3]{216\cdot x^9\cdot y^{15}}[/tex][tex]\sqrt[3]{216\cdot x^9\cdot y^{15}} = \sqrt[3]{6^3\cdot x^9\cdot y^{15}} = 6\cdot x^3\cdot y^5[/tex]The correct option is e. [tex]6\cdot x^3\cdot y^5[/tex]10. [tex]\sqrt{\dfrac{50}{162} } = \sqrt{\dfrac{25\times 2}{81\times 2} } = \sqrt{\dfrac{25}{81} } = \dfrac{5}{9}[/tex]The correct option is c. [tex]\frac{5}{9}[/tex]Learn more about the laws of indices here:https://brainly.com/question/8959311#SPJ1