Use factor method to evaluate the expression below, leaving your answer of a product of its prime factors in power form: sqrt(5184 x 49)

Ili kutatua swali hili, tutatumia njia ya mambo (factor method) kupata mambo makuu ya nambari ndani ya mzizi wa mraba, kisha tutarahisisha.

Sehemu ya 1: Tafuta mambo makuu ya 5184
Tutagawanya 5184 kwa nambari mkuu ndogo zaidi hadi tupate 1.

$$5184 \div 2 = 2592$$ $$2592 \div 2 = 1296$$ $$1296 \div 2 = 648$$ $$648 \div 2 = 324$$ $$324 \div 2 = 162$$ $$162 \div 2 = 81$$

Sasa 81 haigawanyiki kwa 2. Tunajaribu 3.

$$81 \div 3 = 27$$ $$27 \div 3 = 9$$ $$9 \div 3 = 3$$ $$3 \div 3 = 1$$

Kwa hiyo, $5184 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^6 \times 3^4$.

Sehemu ya 2: Tafuta mambo makuu ya 49

$$49 \div 7 = 7$$ $$7 \div 7 = 1$$

Kwa hiyo, $49 = 7 \times 7 = 7^2$.

Sehemu ya 3: Badilisha mambo makuu kwenye usemi
Usemi asili ni $\sqrt{5184 \times 49}$.
Kubadilisha mambo makuu tunapata:

$$\sqrt{(2^6 \times 3^4) \times 7^2}$$

Sehemu ya 4: Rahisisha mzizi wa mraba
Tunatumia kanuni kwamba $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ na $\sqrt{x^n} = x^{n/2}$.

$$\sqrt{2^6 \times 3^4 \times 7^2} = \sqrt{2^6} \times \sqrt{3^4} \times \sqrt{7^2}$$ $$= 2^{6/2} \times 3^{4/2} \times 7^{2/2}$$ $$= 2^3 \times 3^2 \times 7^1$$

Sehemu ya 5: Andika jibu katika mfumo wa nguvu
Jibu tayari liko katika mfumo wa nguvu za mambo makuu.

$$2^3 \times 3^2 \times 7^1$$