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### 2004-D: Dimensional Analysis 1: Simplifying Dimensions

• Topic Cluster: Pure Mathematics
• Topic: Unit Conversion and Dimensional Analysis
• Objective: Simplify theoretical units made of meters, kilograms, and seconds of different powers in the numerator and denominator.
• Content:
• Level: 2

#### BACK to Ladder Unit Conversion and Dimensional Analysis

In this ladder, we are going to learn about how to use dimensional analysis to analyze and confirm physics formulas. Dimensional analysis is a difficult topic, and before we can learn it, we need to practice some mathematical techniques. Firstly, we will practice simplifying fractions made from three units: meters, seconds, and kilograms.

Simplify each of the following combinations of units.

1. $$\frac{\text{m}}{\text{kg}}\cdot\frac{\text{s}}{\text{m}^2}$$
2. $$\frac{\text{m}}{\text{s}^2} \cdot \frac{\text{m}^3}{\text{s}^3} \cdot \frac{\text{kg}^2}{\text{m}^2}$$
3. $$\frac{\text{m}^2}{\text{kg}} \cdot \frac{\text{kg}^3\text{m}}{\text{s}} \cdot \frac{\text{kg}^2\text{s}^3}{\text{m}^3}$$
4. $$\frac{\text{m}^4}{\text{kg}} \cdot \frac{\text{m}}{\text{kg}^2} \cdot \frac{\text{kg}}{\left( \text{m} \cdot \text{s} \right)^3}$$
5. $$\frac{\text{m}^4}{\text{s}^3} \cdot \frac{\text{s}^2}{\text{m}^2} \cdot \frac{\text{kg}}{\text{s}^4}$$
6. $$\frac{\text{m}^5}{\text{kg}^3} \cdot \frac{\text{s}^2}{\text{kg}^2} \cdot \left( \frac{\text{s}}{\text{m} } \right)^2$$
7. $$\frac{\text{s}}{\text{m^3}} \cdot \left( \frac{\text{s}}{\text{m}} \right)^3 \cdot \frac{\text{m}^7}{\text{s}^4}$$
8. $$\frac{\text{s}^4}{\text{kg}^3} \cdot \left( \frac{\text{m}}{\text{kg}} \right)^4 \cdot \left( \frac{\text{s}}{\text{m}} \right)^2$$

### Fractions Made of Fractions

It can be difficult to simplify fractions of fractions. There are two methods by which you can do this:

• Put the denominator of the numerator in the denominator.
• Put the denominator of the denominator in the numerator.

Take whatever fraction is in the denonminator, flip it over, and multiply by it.

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