If x = 64 and y = 27, evaluate: $\frac{x^{\frac{1}{2}} - y^{\frac{1}{3}}}{y - x^{\frac{2}{3}}}$

Step 1: Analyze the question We substitute x = 64 and y = 27 into the expression and simplify, using the laws of indices.

Step 2: Concept

• x^(1/2) is the square root of x.
• y^(1/3) is the cube root of y.
• x^(2/3) = (cube root of x)², i.e. (∛x)².

Step 3: Solution

Numerator:

• x^(1/2) = 64^(1/2) = 8
• y^(1/3) = 27^(1/3) = 3
• Numerator = 8 − 3 = 5

Denominator:

• y = 27
• x^(2/3) = (64^(1/3))² = 4² = 16
• Denominator = 27 − 16 = 11

So the expression = 5/11.

Step 4: Answer

Tip: For fractional indices, take the root first (the denominator of the power), then raise to the power (the numerator). For example, 64^(2/3) = (∛64)² = 4² = 16. This keeps the numbers small and easy to handle without a calculator.