Simplify the Expression: 2^(2x+1) · 2^5 · 2^(3x)

The multiplication rule for exponents states that $a^m \cdot a^n = a^{m+n}$. This comes from the definition of exponents - we're combining repeated multiplication of the same base.

If the bases are different (like $2^3 \cdot 3^2$), you cannot combine them using this rule. The exponent addition rule only works when the bases are identical.

Treat variable terms like regular numbers! Add coefficients: 2x + 3x = 5x, and add constants: 1 + 5 = 6. So 2x+1+5+3x becomes 5x+6.

Absolutely! The rule works for any number of terms with the same base. Just keep adding all the exponents together: $a^m \cdot a^n \cdot a^p \cdot a^q = a^{m+n+p+q}$

The rule still applies! Just add normally, remembering that adding a negative is the same as subtracting. For example: $2^5 \cdot 2^{-2} = 2^{5+(-2)} = 2^3$