Question

5/ Arătaţi că următoarele numere sunt pătrate perfecte:
a 16. 25;
b 218.3°;
e 415. 12117;
f 514.78. 1324;
c25 . 318.
g 325 + 334;
Pls faceți !!!!

Answer 1

Răspuns:

Am notat cu pp = patrat perfect

$\bf a)~16\cdot25=4^2\cdot5^2= \big(4\cdot5\big)^2=\blue{\underline{~20^2=pp}}$

$\bf b)~2^{18}\cdot3^6=2^{9\cdot2}\cdot3^{3\cdot2}=$

$\bf \big(2^9\big)^2\cdot\big(3^3\big)^2=\green{\underline{~\big(2^9\cdot3^3\big)^2=pp~}}$

$\bf c)~25\cdot3^{18}=5^2\cdot3^{9\cdot2}=$

$\bf 5^2\cdot\big(3^9\big)^2=\purple{\underline{~\big(5\cdot 3^9\big)^2=pp~}}$

$\bf d)~16\cdot100=4^2\cdot10^2= \big(4\cdot10\big)^2=\pink{\underline{~40^2~}}$

$\bf e)~4^{15}\cdot121^{17}=\big(2^2\big)^{15}\cdot\big(11^2\big)^{17}=$

$\bf \big(2^{15}\big)^2\cdot\big(11^{17}\big)^2= \red{\underline{~\big(2^{15}\cdot11^{17}\big)^2=pp~}}$

$\bf f)~5^{14}\cdot7^8\cdot13^{24}=5^{7\cdot2}\cdot7^{4\cdot2}\cdot13^{12\cdot2}=$

$\bf \big(5^7\big)^2\cdot\big(7^4\big)^2\cdot\big(13^{12}\big)^2= \blue{\underline{~\big(5^7\cdot7^4\cdot13^{12}\big)^2 = pp~}}$

$\bf g)~3^{25}+3^{24}=3^{24}\cdot\big(3^{25-24}+3^{24-24}\big)=$

$\bf 3^{24}\cdot\big(3^{1}+3^{0}\big)=3^{24}\cdot\big(3+1\big)=$

$\bf 3^{24}\cdot4=\big(3^{17}\big)^2\cdot2^2=\purple{\underline{\big(3^{17}\cdot2\big)^2=pp}}$  

$\bf f)~2^{25}-7\cdot2^{20}=2^{20}\cdot\big(2^{25-20}-7\cdot 2^{20-20}\big)=$

$\bf 2^{20}\cdot\big(2^{5}-7\cdot 2^{0}\big)=2^{20}\cdot\big(32-7\big)=$

$\bf 2^{20}\cdot25=2^{20}\cdot 5^{2} =\red{\underline{~\big(2^{10}\cdot 5\big)^2=pp~}}$

$==pav38==$