Simplify P-6 × P3 × P2 ÷ P4 × P0

Analyze the question: We need to simplify an expression using the laws of indices. The expression is p⁻⁶ × p³ × p² ÷ p⁴ × p⁰ (the "P-6" is p⁻⁶, and so on).

Concept (laws of indices):

• Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ
• Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• Zero power: a⁰ = 1

Step-by-step solution:

p⁻⁶ × p³ × p² ÷ p⁴ × p⁰

Work from left to right, adding the powers for multiplication and subtracting for division:

• p⁻⁶ × p³ = p⁻³
• p⁻³ × p² = p⁻¹
• p⁻¹ ÷ p⁴ = p⁻⁵
• p⁻⁵ × p⁰ = p⁻⁵ (since p⁰ = 1)

Combined in one line: the power is −6 + 3 + 2 − 4 + 0 = −5

So the result is p⁻⁵.

Answer: E.

Tip: Turn the whole expression into one sum of powers: add the powers that are multiplied and subtract the one that is divided. Remember that anything to the power 0 is 1, so it never changes the result. Watch the signs carefully, because a negative power such as p⁻⁶ is where most students slip.