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Tentukanlah nilai x supaya: 4×42×43×...×4142×22×23×...×2x​=1

Pertanyaan

Tentukanlah nilai x supaya:

fraction numerator 2 cross times 2 squared cross times 2 cubed cross times... cross times 2 to the power of x over denominator 4 cross times 4 squared cross times 4 cubed cross times... cross times 4 to the power of 14 end fraction equals 1

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai x yang tepat adalah 20.

Pembahasan

Diketahui:

fraction numerator 2 cross times 2 squared cross times 2 cubed cross times... cross times 2 to the power of x over denominator 4 cross times 4 squared cross times 4 cubed cross times... cross times 4 to the power of 14 end fraction equals 1

Ditanya:

Nilai x

Perlu diingat bahwa:

a to the power of m times a to the power of n equals a to the power of m plus n end exponent open parentheses a to the power of m close parentheses to the power of n equals a to the power of m times n end exponent

Perhatikan perhitungan berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 2 cross times 2 squared cross times 2 cubed cross times... cross times 2 to the power of x over denominator 4 cross times 4 squared cross times 4 cubed cross times... cross times 4 to the power of 14 end fraction end cell equals 1 row cell 2 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end exponent over 4 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus 14 end exponent end cell equals 1 row cell 2 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end exponent over 4 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus 14 end exponent end cell equals 1 row cell 2 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end exponent over open parentheses 2 squared close parentheses to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus 14 end exponent end cell equals 1 row cell 2 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end exponent over 2 to the power of 2 plus 4 plus 6 plus 8 plus 10 plus... plus 28 end exponent end cell equals 1 row cell 2 to the power of 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end exponent end cell equals cell 2 to the power of 2 plus 4 plus 6 plus 8 plus 10 plus... plus 28 end exponent end cell row cell 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end cell equals cell 2 plus 4 plus 6 plus 8 plus 10 plus... plus 28 end cell row cell 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end cell equals cell 2 open parentheses 1 plus 2 plus 3 plus 4 plus 5 plus... plus 14 close parentheses end cell end table

Dengan menggunakan deret aritmatika yaitu S subscript n equals n over 2 open parentheses a plus U subscript n close parentheses, didapat:

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 plus 2 plus 3 plus 4 plus 5 plus... plus x end cell equals cell 2 open parentheses 1 plus 2 plus 3 plus 4 plus 5 plus... plus 14 close parentheses end cell row cell x over 2 open parentheses 1 plus x close parentheses end cell equals cell 2 times 14 over 2 open parentheses 1 plus 14 close parentheses end cell row cell x open parentheses x plus 1 close parentheses end cell equals cell 2 times 14 times 15 end cell row cell x squared plus x end cell equals 420 row cell x squared plus x minus 420 end cell equals 0 row cell open parentheses x plus 21 close parentheses open parentheses x minus 20 close parentheses end cell equals 0 end table
x plus 21 equals 0 space atau space x minus 20 equals 0 x equals negative 21 space atau space x equals 20                                 

Karena nilai x tidak mungkin bernilai negatif, maka nilai x adalah 20.

Jadi, nilai x yang tepat adalah 20.

148

5.0 (4 rating)

Nela Shifa Aulia

Pembahasan tidak lengkap

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