Expand the Expression: 4^(a+b+c) Using Power Properties

The exponent rule states that $x^{m+n} = x^m \times x^n$. This is because exponents represent repeated multiplication. For example: $4^{2+3} = 4^5 = 4 \times 4 \times 4 \times 4 \times 4$, which equals $(4 \times 4) \times (4 \times 4 \times 4) = 4^2 \times 4^3$.

$4^{a+b+c}$ means one base raised to a sum of exponents, which expands to multiplication. $4^a + 4^b + 4^c$ means adding three separate exponential terms, which is completely different and cannot be simplified further.

Yes! This exponent rule works with any base. Whether it's $2^{a+b} = 2^a \times 2^b$, $10^{x+y} = 10^x \times 10^y$, or even $x^{m+n} = x^m \times x^n$ with variables as the base.

Think of it as "same base, add exponents when multiplying" working backwards. If $x^m \times x^n = x^{m+n}$, then $x^{m+n} = x^m \times x^n$. The bases stay the same, and you're just breaking apart the sum in the exponent.

The same rule applies! For example: $4^{a+b+c+d+e} = 4^a \times 4^b \times 4^c \times 4^d \times 4^e$. Just multiply as many terms as you have in the sum.