Pertanyaan

Tentukan nilai a , b , c , dan d dari 2 8 5 ⋅ 9 6 2 ⋅ 62 5 2 ( 3 0 2 ) 4 ⋅ 4 9 6 = 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d

Tentukan nilai $a, b, c, dan d$ dari $\frac{( 3 0 ^{2} ) ^{4} \cdot 4 9 ^{6}}{2 8 ^{5} \cdot 9 6 ^{2} \cdot 62 5 ^{2}} = 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d}$

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E. Lestari

Master Teacher

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Jawaban terverifikasi

Jawaban

nilai a , b , c , dan d berturut-turut adalah a = − 12 , b = 6 , c = 0 dan d = 7 .

nilai

a, b, c, dan d

berturut-turut adalah

a = - 12, b = 6, c = 0 dan d = 7

Pembahasan

Ingat bahwa: ( a b ) c = a b c ( ab ) c = a c b c a c a b = a b − c a b a c = a b + c Mencari nilai a , b , c , dan d dengan mengubah ruas kiri sesuai dengan format pada ruas kanan, didapat: 2 8 5 ⋅ 9 6 2 ⋅ 62 5 2 ( 3 0 2 ) 4 ⋅ 4 9 6 ( 2 2 ⋅ + 4 m u 7 ) 5 ⋅ ( 2 5 ⋅ + 4 m u 3 ) 2 ⋅ ( 5 4 ) 2 3 0 8 ⋅ ( 7 2 ) 6 2 10 ⋅ + 4 m u 2 10 ⋅ + 4 m u 3 2 ⋅ 7 5 ⋅ 5 8 ( 2 ⋅ + 4 m u 3 ⋅ + 4 m u 5 ) 8 ⋅ 7 12 2 10 + 10 ⋅ + 4 m u 3 2 ⋅ + 4 m u 7 5 ⋅ 5 8 2 8 ⋅ + 4 m u 3 8 ⋅ + 4 m u 5 8 ⋅ 7 12 2 20 ⋅ + 4 m u 3 2 ⋅ + 4 m u 7 5 ⋅ 5 8 2 8 ⋅ + 4 m u 3 8 ⋅ + 4 m u 5 8 ⋅ 7 12 2 8 − ( 20 ) ⋅ + 4 m u 3 8 − 2 ⋅ + 4 m u 5 8 − 8 ⋅ 7 12 − 5 2 − 12 ⋅ + 4 m u 3 6 ⋅ + 4 m u 5 0 ⋅ 7 7 = = = = = = = 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d 2 a ⋅ 3 b ⋅ 5 c ⋅ 7 d Karena bentuk tersebut simetri dikedua ruas, sehingga: 2 − 12 − 12 3 6 6 5 0 0 7 7 7 = = = = = = = = 2 a a 3 b b 5 c c 7 d d Jadi,nilai a , b , c , dan d berturut-turut adalah a = − 12 , b = 6 , c = 0 dan d = 7 .

Ingat bahwa:

$(a^{b})^{c} = a^{b c} (ab)^{c} = a^{c} b^{c} \frac{a ^{b}}{a ^{c}} = a^{b - c} a^{b} a^{c} = a^{b + c}$

Mencari nilai $a, b, c, dan d$ dengan mengubah ruas kiri sesuai dengan format pada ruas kanan, didapat:

$\frac{( 3 0 ^{2} ) ^{4} \cdot 4 9 ^{6}}{2 8 ^{5} \cdot 9 6 ^{2} \cdot 62 5 ^{2}} \frac{3 0 ^{8} \cdot ( 7 ^{2} ) ^{6}}{( 2 ^{2} \cdot + 4 m u 7 ) ^{5} \cdot ( 2 ^{5} \cdot + 4 m u 3 ) ^{2} \cdot ( 5 ^{4} ) ^{2}} \frac{( 2 \cdot + 4 m u 3 \cdot + 4 m u 5 ) ^{8} \cdot 7 ^{12}}{2 ^{10} \cdot + 4 m u 2 ^{10} \cdot + 4 m u 3 ^{2} \cdot 7 ^{5} \cdot 5 ^{8}} \frac{2 ^{8} \cdot + 4 m u 3 ^{8} \cdot + 4 m u 5 ^{8} \cdot 7 ^{12}}{2 ^{10 + 10} \cdot + 4 m u 3 ^{2} \cdot + 4 m u 7 ^{5} \cdot 5 ^{8}} \frac{2 ^{8} \cdot + 4 m u 3 ^{8} \cdot + 4 m u 5 ^{8} \cdot 7 ^{12}}{2 ^{20} \cdot + 4 m u 3 ^{2} \cdot + 4 m u 7 ^{5} \cdot 5 ^{8}} 2^{8 - (20)} \cdot + 4 m u 3^{8 - 2} \cdot + 4 m u 5^{8 - 8} \cdot 7^{12 - 5} 2^{- 12} \cdot + 4 m u 3^{6} \cdot + 4 m u 5^{0} \cdot 7^{7} = = = = = = = 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d}$