Question

simplify the expression. \\(\frac{(42m^5n^3)(m^4n^2)}{6m^6n^5}\\) \\(\boxed{}\\)m\\(\boxed{}\\) \\(\\) extra credit: use the given formula to generate the first for terms of each geometric sequence. \\(g_n = 2 \cdot 3^{n - 1}\\) remember... \\(a_n = a_1 \cdot r^{n - 1}\\) n = the term number \\(a_1\\) = the first term in the sequence r = the common ratio \\(\boxed{}\\), \\(\boxed{}\\), \\(\boxed{}\\), \\(\boxed{}\\)

Explanation:

Step1: Simplify coefficients

$\frac{42}{6} = 7$

Step2: Simplify $m$ terms

$m^{5+4-6} = m^{3}$

Step3: Simplify $n$ terms

$n^{3+2-5} = n^{0} = 1$

Step4: Combine simplified parts

$7 \cdot m^3 \cdot 1 = 7m^3$
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Step5: Find 1st geometric term

Substitute $n=1$: $g_1=2 \cdot 3^{1-1}=2 \cdot 3^0=2 \cdot 1=2$

Step6: Find 2nd geometric term

Substitute $n=2$: $g_2=2 \cdot 3^{2-1}=2 \cdot 3^1=2 \cdot 3=6$

Step7: Find 3rd geometric term

Substitute $n=3$: $g_3=2 \cdot 3^{3-1}=2 \cdot 3^2=2 \cdot 9=18$

Step8: Find 4th geometric term

Substitute $n=4$: $g_4=2 \cdot 3^{4-1}=2 \cdot 3^3=2 \cdot 27=54$

Answer:

Simplified expression: $7m^3$ (first blank: 7, second blank: 3)
First four terms of geometric sequence: 2, 6, 18, 54