Si Units

kilogram (kg), metre (m), second (s), ampere (A); kelvin (K)

kilogram; second / kelvin (any two correct)

The kilogram is defined as a base unit by the SI system; the prefix is part of the name only and does not mean it is derived from a smaller unit (the gram).

$$W=Fs$$ $$[W]=({\text{kg m s}}^{-2})(\text{m})$$ $$[W]={\text{kg m}}^{2}{\text{ s}}^{-2}$$

$$P=\frac{W}{t}$$ Units of work = kg m² s⁻² $$[P]=\frac{{\text{kg m}}^{2}{\text{ s}}^{-2}}{\text{s}}$$ $$[P]={\text{kg m}}^{2}{\text{ s}}^{-3}$$ This gives kg m² s⁻³ as required.

$$R=\frac{V}{I}$$ Volt = kg m² s⁻³ A⁻¹ $$[R]=\frac{{\text{kg m}}^{2}{\text{ s}}^{-3}{\text{ A}}^{-1}}{\text{A}}$$ $$[R]={\text{kg m}}^{2}{\text{ s}}^{-3}{\text{ A}}^{-2}$$

An equation in which all terms have the same SI base units.

Left side: energy has base units kg m² s⁻² Right side: $$[m{v}^{2}]=(\text{kg})({\text{m s}}^{-1}{)}^{2}$$ $$[m{v}^{2}]={\text{kg m}}^{2}{\text{ s}}^{-2}$$ Both sides have identical base units, so the equation is homogeneous.

Left side: F = kg m s⁻² Right side: $$\left[\frac{mv}{{t}^{2}}\right]=\frac{(\text{kg})({\text{m s}}^{-1})}{{\text{s}}^{2}}$$ $$={\text{kg m s}}^{-3}$$ The base units differ, so the equation is not homogeneous.

Homogeneity only checks that base units match; it cannot detect missing or wrong dimensionless constants such as ½ or π; an equation may be homogeneous but still numerically wrong.

nano = 10⁻⁹; micro = 10⁻⁶; mega = 10⁶; giga = 10⁹

(a) 4.5 × 10⁹ Hz; (b) 2.5 × 10⁻⁷ m; (c) 6.2 × 10⁻⁶ F

$$470 \mu \text{F}=470\times {10}^{-6}\text{ F}$$ $$=4.70\times {10}^{-4}\text{ F}$$

Prefixes condense very large and very small numbers into manageable forms; this avoids long strings of zeros and reduces transcription errors; for example, atomic separations of order 10⁻¹⁰ m are written as 0.1 nm; capacitor values of 10⁻⁶ F are written as 1 μF; prefixes attach to base or derived units without changing the underlying SI structure; before substitution into equations, prefixed values must be converted back to base SI form to maintain unit consistency throughout calculations.