Question
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Tutor-Verified Answer
Answer
\[ x = 2\sqrt{3} \]
Solution
Berilgan tenglama:
\[ 2^{2x} = 4^{2\sqrt{3}} \]
**YeChish:**
1. Avvalo, \(4\) ni \(2\) ning darajasi sifatida ifodalaymiz:
\[ 4 = 2^2 \]
2. Tenglamada \(4\) ni almashtiramiz:
\[ 2^{2x} = (2^2)^{2\sqrt{3}} \]
3. Darajalarni ko'paytamiz:
\[ 2^{2x} = 2^{4\sqrt{3}} \]
4. Tabiiy asoslar teng bo'lganligi sababli, eksponentlarni tenglashtiramiz:
\[ 2x = 4\sqrt{3} \]
5. Har ikki tomonini \(2\) ga bo'lamiz:
\[ x = 2\sqrt{3} \]
**Natija:**
\[ x = 2\sqrt{3} \]
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Simplify this solution Mind Expander
To solve the equation \( 2^{2x} = 4^{2\sqrt{3}} \), we first rewrite \( 4 \) in terms of base \( 2 \). Since \( 4 = 2^2 \), we have: \[ 4^{2\sqrt{3}} = (2^2)^{2\sqrt{3}} = 2^{4\sqrt{3}}. \] Now, we can substitute this back into the original equation: \[ 2^{2x} = 2^{4\sqrt{3}}. \] Since the bases are the same, we can equate the exponents: \[ 2x = 4\sqrt{3}. \] Now, solving for \( x \) gives us: \[ x = \frac{4\sqrt{3}}{2} = 2\sqrt{3}. \] Thus, the solution to the equation is \( x = 2\sqrt{3} \).
