Ushtrime te zgjidhura te vetive te fuqise. Shumezimi i nje fuqie

Më kemi disa ushtrime me fuqite. Shumëzimin e fuqive me bazë të njëjtë në formë të një fuqie, fuqitë si shumëzim dy fuqish që japin këtë fuqi etj.

Ushtrimet bazohen nga kuptimi i fuqise

Ushtrimi 1

Shkruani shumëzimin e fuqive me bazë të njëjtë në formë të një fuqie:

a) $\displaystyle {{9}^{6}}\cdot {{9}^{8}}=$

b) $\displaystyle {{11}^{4}}\cdot {{11}^{12}}=$

c) $\displaystyle {{6}^{25}}\cdot {{6}^{20}}=$

d) $\displaystyle {{12}^{40}}\cdot {{12}^{30}}=$

Zgjidhje

a) $\displaystyle {{9}^{6}}\cdot {{9}^{8}}={{9}^{6+8}}={{9}^{14}}$ b) $\displaystyle {{11}^{4}}\cdot {{11}^{12}}={{11}^{4+12}}={{11}^{16}}$

c) $\displaystyle {{6}^{25}}\cdot {{6}^{20}}={{6}^{25+20}}={{6}^{45}}$

d) $\displaystyle {{12}^{40}}\cdot {{12}^{30}}={{12}^{40+30}}={{12}^{70}}$

Ushtrimi 2

Të shkruhen fuqitë si shumëzim dy fuqish që japin këtë fuqi:

Për shembull:

$\displaystyle {{6}^{6}}={{6}^{3}}\cdot {{6}^{3}}$

$\displaystyle {{6}^{6}}={{6}^{2}}\cdot {{6}^{4}}$

$\displaystyle {{6}^{6}}={{6}^{1}}\cdot {{6}^{5}}$

$\displaystyle {{6}^{6}}={{6}^{0}}\cdot {{6}^{6}}$

a) $\displaystyle {{8}^{7}}$

b) $\displaystyle {{4}^{5}}$

c) $\displaystyle {{\left( \frac{2}{3} \right)}^{4}}$

d) $\displaystyle {{a}^{10}}$

Zgjidhje:

a) $\displaystyle {{8}^{7}}={{8}^{5}}\cdot {{8}^{2}}$

$\displaystyle {{8}^{7}}={{8}^{6}}\cdot {{8}^{1}}$

$\displaystyle {{8}^{7}}={{8}^{7}}\cdot {{8}^{0}}$

$\displaystyle {{8}^{7}}={{8}^{4}}\cdot {{8}^{3}}$

b) $\displaystyle {{4}^{5}}={{4}^{0}}\cdot {{4}^{5}}$

$\displaystyle {{4}^{5}}={{4}^{1}}\cdot {{4}^{4}}$

$\displaystyle {{4}^{5}}={{4}^{2}}\cdot {{4}^{3}}$

c) $\displaystyle {{\left( \frac{2}{3} \right)}^{4}}={{\left( \frac{2}{3} \right)}^{0}}\cdot {{\left( \frac{2}{3} \right)}^{4}}$

$\displaystyle {{\left( \frac{2}{3} \right)}^{4}}={{\left( \frac{2}{3} \right)}^{1}}\cdot {{\left( \frac{2}{3} \right)}^{3}}$

$\displaystyle {{\left( \frac{2}{3} \right)}^{4}}={{\left( \frac{2}{3} \right)}^{2}}\cdot {{\left( \frac{2}{3} \right)}^{2}}$

d) $\displaystyle {{a}^{10}}={{a}^{5}}\cdot {{a}^{5}}$

$\displaystyle {{a}^{10}}={{a}^{4}}\cdot {{a}^{6}}$

$\displaystyle {{a}^{10}}={{a}^{3}}\cdot {{a}^{7}}$

$\displaystyle {{a}^{10}}={{a}^{2}}\cdot {{a}^{8}}$

$\displaystyle {{a}^{10}}={{a}^{1}}\cdot {{a}^{9}}$

$\displaystyle {{a}^{10}}={{a}^{0}}\cdot {{a}^{10}}$

Ushtrimi 4 (pika a e zgjidhur)

Shkruani herësin e pjestimit të fuqive me bazë të njejtë në formën e një fuqie:

a) $\displaystyle {{4}^{7}}:{{4}^{5}}={{4}^{7-5}}={{4}^{2}}$

b) $\displaystyle {{8}^{12}}:{{8}^{8}}=$

c) $\displaystyle {{10}^{10}}:{{10}^{4}}=$

d) $\displaystyle {{25}^{16}}:{{25}^{9}}=$

e) $\displaystyle {{a}^{50}}:{{a}^{30}}=$

f) $\displaystyle {{x}^{8}}:{{x}^{5}}=$

Ushtrimi 4 (pika a e zgjidhur)

Shkruani fuqitë e dhëna si herës të dy fuqive:

a) $\displaystyle {{7}^{3}}={{7}^{5}}:{{7}^{2}}$

b) $\displaystyle {{10}^{2}}=$

c) $\displaystyle {{25}^{10}}=$

d) $\displaystyle {{a}^{5}}=$

e) $\displaystyle {{x}^{4}}=$

Ushtrimi 5 (pika a e zgjidhur)

Shkruani fuqitë në formën e një fuqie në fuqi:

a) $\displaystyle {{3}^{8}}={{3}^{4}}\cdot {{3}^{2}}$

b) $\displaystyle {{4}^{12}}=$

c) $\displaystyle {{6}^{20}}=$

d) $\displaystyle {{10}^{12}}=$

e) $\displaystyle {{(a+b)}^{24}}=$

f) $\displaystyle {{(x+2)}^{6}}=$

Ushtrimi 6 (pika a e zgjidhur)

Gjeni fuqitë e prodhimeve më poshtë: (zgjidhni mënyrën qe ju duket më e lehtë)

a) Mënyra 1

$\displaystyle {{\left( 3\cdot 6\cdot 4 \right)}^{2}}={{3}^{2}}\cdot {{6}^{2}}\cdot {{4}^{2}}$

$\displaystyle =9\cdot 36\cdot 16$

$\displaystyle =5184$

Mënyra 2

$\displaystyle {{\left( 3\cdot 6\cdot 4 \right)}^{2}}={{72}^{2}}$

= 5184.

b) $\displaystyle {{\left( 5\cdot 4\cdot 6 \right)}^{4}}=$

c) $\displaystyle {{\left( 8\cdot 3\cdot 7 \right)}^{3}}=$

d) $\displaystyle {{\left( 9\cdot 5\cdot a \right)}^{5}}=$

Ushtrimi 7 (pika a dhe b e zgjidhur)

Gjeni vlerën e fuqive me 2 mënyra:

a) Mënyra 1

$\displaystyle {{\left( 3\cdot 4\cdot 7 \right)}^{2}}={{3}^{2}}\cdot {{4}^{2}}\cdot {{7}^{2}}$

$\displaystyle =9\cdot 16\cdot 49$

$\displaystyle =7056$

Mënyra 2

$\displaystyle {{\left( 3\cdot 4\cdot 7 \right)}^{2}}={{74}^{2}}$

$\displaystyle =7056$

a) Mënyra 1

$\displaystyle {{\left( 2\cdot 6\cdot 5 \right)}^{3}}={{2}^{3}}\cdot {{6}^{3}}\cdot {{5}^{3}}$

$\displaystyle =8\cdot 216\cdot 125$

$\displaystyle =216'000$

Mënyra 2

$\displaystyle {{\left( 2\cdot 6\cdot 5 \right)}^{3}}={{60}^{3}}$

$\displaystyle =216'000$

c) $\displaystyle {{\left( 5\cdot 3\cdot 2 \right)}^{3}}=$

d) $\displaystyle {{\left( 6\cdot 8\cdot 9 \right)}^{4}}=$

e) $\displaystyle {{\left( 3\cdot a\cdot b \right)}^{3}}=$

f) $\displaystyle {{\left[ \left( -\frac{2}{3} \right)\cdot \left( -3\frac{4}{7} \right)\cdot \left( +3 \right) \right]}^{2}}=$

Ushtrimi 8 (I pazgjidhur)

Njehsoni me dy mënyra:

a) $\displaystyle {{\left( 8:4 \right)}^{2}}=$

b) $\displaystyle {{\left( 9:3 \right)}^{3}}=$

c) $\displaystyle {{\left( 32:2 \right)}^{5}}=$

d) $\displaystyle {{\left( 3a:5b \right)}^{4}}=$

e) $\displaystyle {{\left[ \left( -3\frac{1}{2} \right):5 \right]}^{2}}=$

f) $\displaystyle {{\left[ \left( -9\frac{2}{3} \right):\left( -4 \right) \right]}^{3}}=$

Ushtrimi 9 (pika a e zgjidhur)

Njehsoni vlerat e shprehjes me dy mënyra:

a) Mënyra 1

$\displaystyle {{49}^{2}}:{{7}^{2}}={{\left( 49:7 \right)}^{2}}$ .

$\displaystyle ={{7}^{2}}$

$\displaystyle =49$

Mënyra 2

$\displaystyle {{49}^{2}}:{{7}^{2}}=2401:49$

$\displaystyle =49$

b) $\displaystyle {{125}^{3}}:{{25}^{3}}=$

c) $\displaystyle {{64}^{3}}:{{4}^{3}}=$

d) $\displaystyle {{\left( -4.5 \right)}^{2}}:{{\left( -1.5 \right)}^{2}}=$

Ushtrimi 10 (pika a e zgjidhur)

Njehsoni vlerat e shprehjeve. Aty ku s’ka mundësi, paraqiteni në formë të një fuqie:

a) $\displaystyle {{\left( 2\cdot 3\cdot 5 \right)}^{2}}+{{20}^{2}}:{{4}^{2}}=$

$\displaystyle {{30}^{2}}+{{\left( 20:4 \right)}^{2}}=$

$\displaystyle {{30}^{2}}+{{5}^{2}}=$

$\displaystyle 900+25=$

$\displaystyle =925$

b) $\displaystyle {{\left( 2\cdot 4\cdot 6 \right)}^{3}}:{{5}^{3}}:{{3}^{3}}=$

c) $\displaystyle {{\left( 5\cdot 2\cdot 4 \right)}^{2}}-{{6}^{2}}:{{2}^{2}}=$

d) $\displaystyle {{\left( x+a \right)}^{2}}\cdot {{\left( x+a \right)}^{2}}=$