Without using mathematical tables or calculator: (81/16)3/4 × (27/64)⁻²/3

Question

ScanSolve AI Tutor · Accepted Answer

Huu hapa ni ufumbuzi wa swali lako la 5:

Swali la 5: Bila kutumia jedwali za hisabati au kikokotoo, tathmini:

\left(\frac{81}{16}\right)^{3/4} \times \left(\frac{27}{64}\right)^{-2/3}

Rahisisha sehemu ya kwanza ya usemi.
Tunajua kuwa $81 = 3^4$ na $16 = 2^4$ .
$\left(\frac{81}{16}\right)^{3/4} = \left(\frac{3^4}{2^4}\right)^{3/4} = \left(\left(\frac{3}{2}\right)^4\right)^{3/4}$
Tumia sheria ya vipeo $(a^m)^n = a^{mn}$ :
$\left(\frac{3}{2}\right)^{4 \times \frac{3}{4}} = \left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$
Rahisisha sehemu ya pili ya usemi.
Tunajua kuwa $27 = 3^3$ na $64 = 4^3$ .
$\left(\frac{27}{64}\right)^{-2/3} = \left(\frac{3^3}{4^3}\right)^{-2/3} = \left(\left(\frac{3}{4}\right)^3\right)^{-2/3}$
Tumia sheria ya vipeo $(a^m)^n = a^{mn}$ :
$\left(\frac{3}{4}\right)^{3 \times \left(-\frac{2}{3}\right)} = \left(\frac{3}{4}\right)^{-2}$
Tumia sheria ya vipeo hasi $a^{-n} = \frac{1}{a^n} = \left(\frac{1}{a}\right)^n$ :
$\left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}$
Zidisha matokeo ya sehemu zote mbili.
$\frac{27}{8} \times \frac{16}{9}$
Rahisisha kwa kufuta namba zinazofanana:
$\frac{27}{9} \times \frac{16}{8} = 3 \times 2 = 6$
Jibu la mwisho ni 6.

Without using mathematical tables or calculator: (81/16)3/4 * (27/64)⁻²/3