Power Play — Practice

Question 1

Simplify: 7³ × 7⁵

Accepted Answer

✦ Answer: Using Law 1 (product rule): 7³ × 7⁵ = 7³⁺⁵ = 7⁸. We KEEP the base 7 and ADD the exponents.

Question 2

Evaluate 3⁻²

Accepted Answer

✦ Answer: 3⁻² = 1/3² = 1/9. Negative exponent means reciprocal: 1 divided by the positive power.

Question 3

Simplify: (2⁻³ × 2⁵) ÷ 2⁻⁴

Accepted Answer

Step 1 — Numerator (Law 1).
  2⁻³ × 2⁵ = 2⁻³⁺⁵ = 2²

Step 2 — Division (Law 2).
  2² ÷ 2⁻⁴ = 2²⁻⁽⁻⁴⁾ = 2²⁺⁴ = 2⁶

Step 3 — Evaluate.
  2⁶ = 64

✦ Answer: 64

Question 4

Express the speed of light (299,792,458 m/s) in standard form to 3 significant figures.

Accepted Answer

Step 1 — Identify the number.
  299,792,458 m/s

Step 2 — Round to 3 significant figures.
  299,792,458 ≈ 300,000,000 m/s
  (Actually 3.00 × 10⁸ — the 0s are kept for sig fig display)

Step 3 — Convert to standard form.
  Move decimal LEFT to get 3.00 (after the 3, 8 places moved)
  Standard form: 3.00 × 10⁸ m/s

Step 4 — More precise form.
  Actually 2.998 × 10⁸ m/s (to 4 sig fig)

✦ Answer: The speed of light is approximately 3 × 10⁸ m/s (or 2.998 × 10⁸ m/s precisely). This is one of the most fundamental constants in physics.

Question 5

If a virus has a diameter of 1.2 × 10⁻⁷ m, and 1 cm of a microscope slide can fit viruses side by side, how many viruses fit in 1 cm?

Accepted Answer

Step 1 — Set up the problem.
  Length of slide: 1 cm = 1 × 10⁻² m = 10⁻² m
  Diameter of one virus: 1.2 × 10⁻⁷ m

Step 2 — Compute the number.
  Number = (Length of slide) ÷ (Diameter of one virus)
        = (1 × 10⁻²) ÷ (1.2 × 10⁻⁷)

Step 3 — Apply Law 2 (divide).
  = (1 ÷ 1.2) × 10⁻²⁻⁽⁻⁷⁾
  = 0.833... × 10⁵
  = 8.33 × 10⁴

Step 4 — Convert to integer.
  8.33 × 10⁴ = 83,333 viruses (approximately)

Step 5 — Verify magnitude.
  Slide is 10⁻² m; virus is 10⁻⁷ m; ratio is 10⁵. So about 100,000 viruses fit in 1 cm. Our 83,333 is in this range — sensible.

✦ Answer: Approximately 83,333 viruses fit side-by-side in 1 cm. This shows how SMALL viruses are — we can fit over 80,000 in just 1 cm.

Practice — Power Play

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