When multiplying exponents with the same base, add the powers.

Question

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Here are the solutions to the problems.

2. Simplify the expression: $C^4 \times C^3 \times C^5$

When multiplying exponents with the same base, add the powers.
$C^{4+3+5}$
Add the exponents.
$C^{12}$
The simplified expression is $\boxed{C^{12}}$ .

3. Simplify the expression: $2^5 \times 3^7 \times 2^{11} \times 3^6$
Group terms with the same base.
$(2^5 \times 2^{11}) \times (3^7 \times 3^6)$
Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) to each base.
$2^{5+11} \times 3^{7+6}$
Add the exponents for each base.
$2^{16} \times 3^{13}$
The simplified expression is $\boxed{2^{16} \times 3^{13}}$ .

4. Simplify the expression: $5m^2n \times 12mn^3$
Multiply the coefficients.
$5 \times 12 = 60$
Multiply the terms with base $m$ .
Remember $m = m^1$ .
$m^2 \times m^1 = m^{2+1} = m^3$
Multiply the terms with base $n$ .
Remember $n = n^1$ .
$n^1 \times n^3 = n^{1+3} = n^4$
Combine the results.
$60m^3n^4$
The simplified expression is $\boxed{60m^3n^4}$ .

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