Question

1)( 25^3/49) умножить на (14^2/10^6) 2) 7^40+ 7^38-2*7^39 _______________ 6^2* 49^19 3) (36^3/18^4 )*(15^2/10^3) 4) 3^48-3^47+17*3^46 ________________ 27^15 * 23

Answer 1

$1)\; \; \frac{25^3}{49}\cdot \frac{14^2}{10^6}=\frac{5^6\; \cdot \; (2^2\cdot 7^2)}{7^2\; \cdot \; (2^6\cdot 5^6)}=\frac{1}{2^4}=\frac{1}{16}\\\\\\2)\; \; \frac{7^{40}+7^{38}-2\cdot 7^{39}}{6^2\cdot 49^{19}}=\frac{7^{38}\cdot \; (7^2+1-2\cdot 7)}{36\cdot 7^{38}}=\frac{49+1-14}{36}=\frac{36}{36}=1$

$3)\; \; \frac{36^3}{18^4}\cdot \frac{15^2}{10^3}=\frac{(2^2\cdot 3^2)^3\; \cdot \; (3\cdot 5)^2}{(2\cdot 3^2)^4\; \cdot \; (2\cdot 5)^3}=\frac{2^6\cdot 3^6\; \cdot 3^2\cdot 5^2}{2^4\cdot 3^8\cdot 2^3\cdot 5^3}=\frac{2^6\cdot 3^8\cdot 5^2}{2^7\cdot 3^8\cdot 5^3}=\frac{1}{2\cdot 5}=\frac{1}{10}=0,1\\\\\\4)\; \; \frac{3^{48}-3^{47}+17\cdot 3^{46}}{27^{15}\cdot 23}=\frac{3^{46}\cdot \; (3^2-3+17)}{(3^3)^{15}\cdot 23}=\frac{3^{46}\cdot 23}{3^{45}\cdot 23}=\frac{3\cdot \; 23}{23}=3$

Answer 2

Let's solve each of these mathematical expressions step-by-step:

1. $\frac{{25^3}}{{49}} \times \frac{{14^2}}{{10^6}}$

Step 1: Simplify each fraction. $\frac{{25^3}}{{49}} = \frac{{15625}}{{49}}$ $\frac{{14^2}}{{10^6}} = \frac{{196}}{{1000000}}$

Step 2: Multiply the two fractions. $\frac{{15625}}{{49}} \times \frac{{196}}{{1000000}} = \frac{{15625 \times 196}}{{49 \times 1000000}}$

Step 3: Perform the multiplication. $\frac{{15625 \times 196}}{{49 \times 1000000}} = \frac{{3053125}}{{49000000}} = \frac{{61}}{{1220}}$

2. $\frac{{7^{40} + 7^{38} - 2 \times 7^{39}}}{{6^2 \times 49^{19}}}$

Step 1: Simplify the numerator. $7^{40} + 7^{38} - 2 \times 7^{39} = 7^{38} \times (7^2 + 1 - 2 \times 7)$

Step 2: Perform the operations in the parentheses. $7^2 + 1 - 2 \times 7 = 49 + 1 - 14 = 36$

Step 3: Substitute back into the expression. $7^{38} \times 36$

Step 4: Simplify further. $7^{38} \times 36 = \frac{{7^{38} \times 36}}{{1}} = \frac{{252}}{{1}} = 252$

Step 5: Simplify the denominator. $6^2 \times 49^{19} = 36 \times 49^{19}$

Step 6: The final result. $\frac{{252}}{{36 \times 49^{19}}}$

3. $\frac{{36^3}}{{18^4}} \times \frac{{15^2}}{{10^3}}$

Step 1: Simplify each fraction. $\frac{{36^3}}{{18^4}} = \frac{{46656}}{{104976}}$ $\frac{{15^2}}{{10^3}} = \frac{{225}}{{1000}}$

Step 2: Multiply the two fractions. $\frac{{46656}}{{104976}} \times \frac{{225}}{{1000}} = \frac{{46656 \times 225}}{{104976 \times 1000}}$

Step 3: Perform the multiplication. $\frac{{46656 \times 225}}{{104976 \times 1000}} = \frac{{10497600}}{{104976 \times 1000}}$

Step 4: Simplify further. $\frac{{10497600}}{{104976 \times 1000}} = \frac{{10000}}{{1000}} = 10$