Simplify 25$^{(\frac{1}{2}}$ × 8$^{(\frac{-2}{3}}$

Analyze the question: We need to simplify 25^(1/2) × 8^(-2/3). This uses fractional and negative indices.

Concept:

• a^(1/n) means the nth root of a.
• a^(-m) = 1/a^m
• a^(m/n) = (ⁿ√a)^m

Solution:

Step 1: 25^(1/2) = √25 = 5

Step 2: 8^(-2/3) = 1 / 8^(2/3)

• The cube root of 8 is 2
• 2² = 4
• So 8^(2/3) = 4, and 8^(-2/3) = 1/4

Step 3: Multiply: 5 × 1/4 = 5/4 = 1 1/4

Tip: For fractional powers, take the root first, then the power. This keeps the numbers small. For negative powers, flip to the reciprocal first. Remember that 8, 27, 64 are perfect cubes, and 4, 9, 25, 36 are perfect squares, which come up often in JAMB index questions.