show that the famous 'Einstein equation' E=mc^2 is dimentionally consistent?

The fundamental units are mass, length & time:

E = F*L = m*a*L = m*(L/t²)*L = m*(L²/t²)

mc² = m*(L²/t²)

QED

show that the famous 'Einstein equation' E=mc^2 is dimentionally consistent?

E = mass * c^2

E = energy

m = mass = kg

c = speed of light = 3 * 10^8 m/s

c^2 = m^2 / s^2

mc^2 = kg * m^2/s^2

The unit of energy = Joule = Newton * meters

Newton is unit of Force, F = mass * a

Mass = kg, a = m/s^2, so F = kg * m /s^2

Joule = Newton * m = kg * m /s^2 * m

Joule = kg * m^2 /s^2

mc^2 = kg * m^2/s^2

Units agree

Since the question is whether the equation is dimensionally correct? we need not analyse it dimensionally. It is enough, if the left and right side of this equation represents the same physical quantity.

L.H.S is energy.

R.H.S is mass times square of velocity which is kinetic energy and hence energy.There are no other quantity in the right side other than energy.

Hence it is dimensionally correct.

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sure, same as kinetic energy = 1/2 mv^2

m is in kg

v is in m/s

so mc^2 has the same units as 1/2 mv^2 Both are in kg*m^2/s^2 which is a Joule, the standard unit of energy (E)

The unit of energy = Joules =Nm

As,Energy =F×L

Dimensions of l.h.s of eq.= [E]= [ML^2 T^-2]

Dimensions of r.h.s of the eq. = [ML^2 T ^-2]

SINCE lhs = rhs

Eq is dimensionally consistent

Use dimensional analysis and the base units for M, L, T.