Math Snap

STEP 1

1. The variable $y$ represents a positive real number.
2. We need to simplify the expression by simplifying the square roots and combining like terms if possible.

STEP 2

1. Simplify the square root expressions individually.
2. Combine the simplified expressions.

STEP 3

Simplify the first square root expression $\sqrt{48 y^3}$.
First, factor $48$ into its prime factors:
$$48 = 2^4 \times 3$$ Rewrite the expression under the square root:
$$\sqrt{48 y^3} = \sqrt{2^4 \times 3 \times y^3}$$ Separate the perfect squares from the non-perfect squares:
$$= \sqrt{(2^2)^2 \times 3 \times y^2 \times y}$$ Take the square root of the perfect squares:
$$= 2^2 \times y \times \sqrt{3y}$$ $$= 4y \sqrt{3y}$$

STEP 4

Simplify the second square root expression $\sqrt{27 y^5}$.
First, factor $27$ into its prime factors:
$$27 = 3^3$$ Rewrite the expression under the square root:
$$\sqrt{27 y^5} = \sqrt{3^3 \times y^5}$$ Separate the perfect squares from the non-perfect squares:
$$= \sqrt{3^2 \times 3 \times (y^2)^2 \times y}$$ Take the square root of the perfect squares:
$$= 3 \times y^2 \times \sqrt{3y}$$ $$= 3y^2 \sqrt{3y}$$

STEP 5

Substitute the simplified square root expressions back into the original expression:
$$5y \times 4y \sqrt{3y} - 3y^2 \sqrt{3y}$$ $$= 20y^2 \sqrt{3y} - 3y^2 \sqrt{3y}$$

Combine like terms:
$$= (20y^2 - 3y^2) \sqrt{3y}$$ $$= 17y^2 \sqrt{3y}$$ The simplified expression is:
$$\boxed{17y^2 \sqrt{3y}}$$