Explicație pas cu pas:
c)
[tex]= \frac{ {4}^{n} \cdot {5}^{n + 1} + {4}^{n + 1} \cdot {5}^{n}}{{3}^{n} \cdot {7}^{n + 1} + {3}^{n + 1} \cdot {7}^{n} - {21}^{n} } = \\ [/tex]
[tex]= \frac{ {4}^{n} \cdot {5}^{n} \cdot 5 + {4}^{n} \cdot 4 \cdot {5}^{n}}{{3}^{n} \cdot {7}^{n} \cdot 7 + {3}^{n} \cdot 3 \cdot {7}^{n} - {3}^{n} \cdot {7}^{n}} \\ [/tex]
[tex]= \frac{{4}^{n} \cdot {5}^{n}(5 + 4)}{{3}^{n} \cdot {7}^{n}(7 + 3 - 1)} = \frac{{4}^{n} \cdot {5}^{n} \cdot \red{\bf 9}}{{3}^{n} \cdot {7}^{n} \cdot \red{\bf 9}} \\ [/tex]
d)
[tex]\frac{{6}^{n + 2} + {6}^{n}}{{5}^{n + 2} - 6 \cdot {5}^{n}} = \frac{{6}^{n}({6}^{2} + 1)}{{5}^{n}( {5}^{2} - 6)} = \frac{{6}^{n}(36 + 1)}{{5}^{n}(25 - 6)} = \frac{{6}^{n} \cdot 37}{{5}^{n} \cdot 19} \\ [/tex]
=> fracție ireductibilă
e)
[tex]\frac{\overline {a10} + \overline {b34} + \overline {c56} }{\overline {d27} + \overline {e32} + \overline {f41}} = \\ [/tex]
[tex]= \frac{100a + 10 + 100b + 34 + 100c + 56}{100d + 27 + 100e + 32 + 100f + 41}\\[/tex]
[tex]= \frac{100(a + b + c) + 100}{100(d + e + f) + 100} \\ [/tex]
[tex]= \frac{\red{\bf100} \cdot (a + b + c + 1)}{\red{\bf100} \cdot (d + e + f + 1)} \\ [/tex]
