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Negative Index Laws Proofs

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Negative Index Laws Proofs

Show your reasoning for proof and simplification questions. Assume any variable used as a denominator is non-zero.

Review: Positive Index Laws

1.Which index law justifies changing p⁴ × p⁷ into p¹¹?

Multiply powers with the same base by adding the indices: pᵐ × pⁿ = pᵐ⁺ⁿ
Divide powers with the same base by subtracting the indices: pᵐ ÷ pⁿ = pᵐ⁻ⁿ
Raise a power to a power by multiplying the indices: (pᵐ)ⁿ = pᵐⁿ
Multiply the base and index: pᵐ × pⁿ = (p × m) + (p × n)

2.Simplify 7⁵ ÷ 7⁵. State the index law used.

3.Use the division law to simplify r⁶ ÷ r⁹ as a single power of r.

4.Match each expression to the index law it demonstrates.

q³ × q⁸ = q¹¹
t⁹ ÷ t⁴ = t⁵
(u²)⁶ = u¹²
v⁰ = 1 (v ≠ 0)

Divide same-base powers
Zero index law
Raise a power to a power
Multiply same-base powers

Build a Proof for Negative Indices

Use the positive index laws to explain why a negative index represents a reciprocal.

5.Complete the reasoning for a⁻ᵏ = 1/aᵏ, where a ≠ 0. Fill each blank: (i) aᵏ ÷ aᵏ = __; (ii) by the division law, aᵏ ÷ aᵏ = aᵏ⁻ᵏ = a__; (iii) therefore a⁰ = __; (iv) a⁰ ÷ aᵏ = 1 ÷ aᵏ = __; (v) extending the division law consistently assigns the index 0 − k to this quotient, so define a⁰⁻ᵏ as __; (vi) therefore a⁻ᵏ = __.

6.Use the negative index definition to show why 5⁻² equals 1/25. Include the reciprocal step and evaluate the positive power.

7.Starting with the power law (xᵐ)ⁿ = xᵐⁿ, prove that (x⁻³)² = x⁻⁶. State the values of the indices you substitute.

Apply and Check

8.Rewrite 6⁻³ as a fraction with a positive index, then evaluate it.

9.Simplify z⁵ × z⁻⁸ as a single power of z.

10.Rewrite (3w)⁻² using positive indices and simplify the numerical factor.

11.Select every statement that is true for non-zero variables.

8⁻¹ = −8
8⁻¹ = 1/8
m⁻⁴ × m⁴ = 1
(cd)⁻² = 1/(c²d²)
(y⁻²)³ = y⁻⁵

12.Find and correct the error in this working: 4⁻² = −4² = −16.

13.Simplify (a⁻² × a⁵) ÷ a⁻¹ as a single power of a, then write the result using positive indices.

14.Write a brief proof outline explaining why 3⁻⁴ = 1/3⁴. Show that 3⁰ ÷ 3⁴ = 1/3⁴, then explain how extending the division law consistently defines 3⁻⁴.