Problems from Operations with powers

Express with powers the following expressions:

  1. $$\displaystyle \frac{7^3 \cdot 3^{14} \cdot (7^2)^12}{3^5\cdot 7}$$

  2. $$\displaystyle \frac{(4^4)^7 \cdot 5^2\cdot 4^5 \cdot 5^{24}}{5^{11}}$$
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Development:

  1. $$\displaystyle \frac{7^3 \cdot 3^{14} \cdot (7^2)^{12}}{3^5\cdot 7}=\frac{7^3\cdot 3^{14}\cdot 7^{2 \cdot 12}}{3^5\cdot 7}=\frac{7^3\cdot 3^{14} \cdot 7^{24}}{3^5\cdot 7}=$$

    $$=7^{3+24} \cdot 3^{14-5}=7^{26}\cdot 3^9$$

  2. $$\displaystyle \frac{(4^4)^7\cdot 5^2\cdot 4^5\cdot 5^{24}}{5^{11}}= \frac{4^{4 \cdot 7}5^2\cdot 4^5\cdot 5^{24}}{5^{11}}=\frac{5^{2+24} \cdot 4^{28+5}}{5^{11}}=$$

    $$=5^{26-11}\cdot 4^{33}= 5^{15}\cdot4^{33}$$

Solution:

  1. $$7^{26}\cdot 3^{9}$$
  2. $$5^{15}\cdot 4^{33}$$
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