MCQ
$\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{81}{16}$ then $x =$
- A$2$
- B$3$
- ✓$5$
- D$1$
✓
Answer
Correct option: C.
$5$
We have to find value of x provided $\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{81}{16}$
So,
$\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{81}{16}$
$\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{3^4}{2^4}$
$\frac{2^\text{x}}{3^\text{x}}\frac{3^{2\text{x}}}{2^{2\text{x}}}=\frac{3^4}{2^4}$
$\frac{3^{2\text{x}-\text{x}}}{2^{2\text{x}-\text{x}}}=\frac{3^4}{2^4}$
$\frac{3^\text{x}}{2^\text{x}}=\frac{3^4}{2^4}$
Equating exponents of power we get $x = 5$
Hence the correct alternative is $c.$
So,
$\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{81}{16}$
$\Big(\frac{2}{3}\Big)^\text{x}\Big(\frac{3}{2}\Big)^{2\text{x}}=\frac{3^4}{2^4}$
$\frac{2^\text{x}}{3^\text{x}}\frac{3^{2\text{x}}}{2^{2\text{x}}}=\frac{3^4}{2^4}$
$\frac{3^{2\text{x}-\text{x}}}{2^{2\text{x}-\text{x}}}=\frac{3^4}{2^4}$
$\frac{3^\text{x}}{2^\text{x}}=\frac{3^4}{2^4}$
Equating exponents of power we get $x = 5$
Hence the correct alternative is $c.$
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