Final answer:
The expression with square roots simplifies to x squared times y to the 4.5th power, or x squared times y to the 4th power times the square root of y, applying the rules for multiplying exponents and expressing square roots as fractional powers.
Step-by-step explanation:
To solve the expression involving square roots and variables raised to different powers, we apply the rules of exponents and radical expressions. The student asks about the square root of xy2 multiplied by the square root of x3y7. First, we express the square roots as fractional powers, and then we use the rule of multiplication of exponents.
For square roots, the rule is that the square root of x can be rewritten as x raised to the 1/2 power. As in the examples provided, the rule for raising a number to a power (xp) and then multiplying it by the same base raised to another power (xq) is to add the exponents, resulting in xp+q. Finally, the cube of an exponential involves cubing the digit term and multiplying the exponent by 3.
In this case:
- Square root of xy2 is x1/2y
- Square root of x3y7 is x3/2y7/2
When we multiply these together:
(x1/2y) * (x3/2y7/2) = x1/2+3/2y1+7/2 = x2y9/2
This expression simplifies to x2y4.5 or x2y4√y.