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Difficulty: MediumExponents, Roots, and Scientific Notation

A high-speed laser pulse travels at a constant speed of 3.0×1083.0 \times 10^8 meters per second. If the pulse takes 4.5×10−74.5 \times 10^{-7} seconds to travel the entire length of a specialized optical fiber, what is the length of the optical fiber, in meters?

Answer: 135 meters

Answer

The length of the optical fiber is 135 meters.
Multiplying the speed 3.0×1083.0 \times 10^8 meters per second by the time 4.5×10−74.5 \times 10^{-7} seconds yields (3.0×4.5)×108+(−7)=13.5×101=135(3.0 \times 4.5) \times 10^{8 + (-7)} = 13.5 \times 10^1 = 135 meters.

Step-by-Step Solution

1
Set up the distance equation
d=(3.0×108)×(4.5×10−7)d = (3.0 \times 10^8) \times (4.5 \times 10^{-7})
Distance equals speed multiplied by time.
2
Group coefficients and powers of 10
d=(3.0×4.5)×(108×10−7)d = (3.0 \times 4.5) \times (10^8 \times 10^{-7})
The commutative and associative properties of multiplication allow grouping coefficients and base-10 exponential terms together.
3
Simplify the coefficients and add exponents
d=13.5×101=135d = 13.5 \times 10^1 = 135
Multiplying the coefficients yields 13.5, and adding the exponents gives 8+(−7)=18 + (-7) = 1. Evaluating 13.5×10113.5 \times 10^1 results in 135.

Key Concept

Multiplying numbers expressed in scientific notation using exponent rules.
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Topics

Exponents, Roots, and Scientific Notation

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