Product Rule for Exponents

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When multiplying powers with the same base, you add the exponents:

[ a^m \cdot a^n = a^{m+n} ]

This works because each factor of the base is being counted together in one product.

Make sure the bases are identical. The rule only applies when the repeated factor is the same number or variable.

Keep the base and combine the powers by adding the exponents. For example:

[ x^3 \cdot x^5 = x^{3+5} = x^8 ]

If there are more than two factors, add all the exponents:

[ a^2 \cdot a^4 \cdot a^1 = a^{2+4+1} = a^7 ]

Write the answer in simplest exponent form. Do not multiply the exponents.

You can expand a small example to verify the rule. For instance,

[ x^2 \cdot x^3 = (x\cdot x)(x\cdot x\cdot x)=x^5 ]

So the exponent should be the sum of the original exponents.

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