Question - Simplified Expression with Square Roots and Cubes
Solution:
The expression given in the image is:$$\frac{2y\sqrt{x^3} + y^3 - 2x \times 2y}{2\sqrt{x^3} + y^3} \div \sqrt{x^3 + y^3}$$Let's simplify it step by step.First, we can simplify the numerator by combining like terms:\[2y\sqrt{x^3} + y^3 - 2x \times 2y = 2y\sqrt{x^3} + y^3 - 4xy\]Now, we recognize that $$\sqrt{x^3}$$ is $$x^{3/2}$$. This simplifies the expression to:$$\frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \div \sqrt{x^3 + y^3}$$Next, let's simplify the divisor. $$\sqrt{x^3 + y^3}$$ is the root of the sum of cubes, and it is not factorable in the same simple way as the sum of squares. So, we leave it as is.Now we can rewrite the division as multiplication by the reciprocal:$$\frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \times \frac{1}{\sqrt{x^3 + y^3}}$$Now we multiply the fractions:$$\frac{(2yx^{3/2} + y^3 - 4xy) \times 1}{(2x^{3/2} + y^3) \times \sqrt{x^3 + y^3}}$$Since there are no like terms and no obvious factorizations to simplify further, and we're not given values for $$x$$ or $$y$$, this seems to be as simplified as we can get without further instructions or context.Therefore, the simplified form of the given expression is:$$\frac{2yx^{3/2} + y^3 - 4xy}{(2x^{3/2} + y^3) \sqrt{x^3 + y^3}}$$
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