Question

Rewrite the expression with rational exponents as a radical expression by extending the properties of integer exponents.

y to the three fourths power, all over y to the one half power

the eighth root of y to the third power
the square root of y to the three fourths power
the fourth root of y
the square root of y

Answer 1

y^(3/4) / y^(1/2) can be rewritten as y^((3/4)-(1/2)) using the quotient of powers property of exponents, which gives us:

y^(3/4 - 1/2) = y^(3/4 - 2/4) = y^(1/4)

So, y^(3/4) / y^(1/2) = y^(1/4)

The expression (y^3)^(1/8) can be rewritten as y^(3/8) using the power of a power property of exponents, which gives us:

(y^3)^(1/8) = y^(3/8)

Then, we can apply the radical notation to obtain:

the eighth root of y to the third power is ∛(y^3) = y^(3/8)

The expression (y^(3/4))^(1/2) can be rewritten as y^((3/4)*(1/2)) using the power of a power property of exponents, which gives us:

(y^(3/4))^(1/2) = y^(3/4 * 1/2) = y^(3/8)

Then, we can apply the radical notation to obtain:

the square root of y to the three fourths power is √(y^(3/4)) = y^(3/8)

The expression y^(1/4) can be rewritten as the fourth root of y, since taking the fourth root of y is equivalent to raising y to the power of 1/4:

the fourth root of y is y^(1/4)

The expression y^(1/2) can be rewritten as the square root of y, since taking the square root of y is equivalent to raising y to the power of 1/2:

the square root of y is y^(1/2)