Bentuk sederhana dari 3 2/3 × 8 2/3 / 22 5/7 × 96 5/6 adalah ....

Question

ScanSolve AI Tutor · Accepted Answer

To simplify the expression, we will convert all bases to their prime factors and then apply the rules of exponents.

Express all bases as powers of prime numbers.
We have $8 = 2^3$ and $9 = 3^2$ .
Substitute these prime factorizations into the expression.
$\frac{3^{\frac{2}{3}} \times (2^3)^{\frac{3}{2}}}{2^{\frac{5}{2}} \times (3^2)^{\frac{5}{6}}}$
Apply the power of a power rule $(a^m)^n = a^{mn}$ .
$\frac{3^{\frac{2}{3}} \times 2^{3 \times \frac{3}{2}}}{2^{\frac{5}{2}} \times 3^{2 \times \frac{5}{6}}}$ $\frac{3^{\frac{2}{3}} \times 2^{\frac{9}{2}}}{2^{\frac{5}{2}} \times 3^{\frac{10}{6}}}$
Simplify the exponent $\frac{10}{6}$ to $\frac{5}{3}$ .
$\frac{3^{\frac{2}{3}} \times 2^{\frac{9}{2}}}{2^{\frac{5}{2}} \times 3^{\frac{5}{3}}}$
Group terms with the same base and apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ .
For base 2:
$2^{\frac{9}{2} - \frac{5}{2}} = 2^{\frac{9-5}{2}} = 2^{\frac{4}{2}} = 2^2$
For base 3:
$3^{\frac{2}{3} - \frac{5}{3}} = 3^{\frac{2-5}{3}} = 3^{\frac{-3}{3}} = 3^{-1}$
Combine the simplified terms and calculate the final value.
$2^2 \times 3^{-1} = 4 \times \frac{1}{3} = \frac{4}{3}$
The simplified form of the expression is $\frac{4}{3}$ .

Comparing this with the given options:
A. $\frac{1}{42}$
B. $\frac{2}{3}$
C. $\frac{4}{3}$
D. 6
E. 12

The correct option is C.

The final answer is $\boxed{\text{C. } \frac{4}{3}}$ .

Bentuk sederhana dari 3 2/3 x 8 2/3 / 22 5/7 x 96 5/6 adalah ....