Let's solve the equation step by step:

Given equation: $\sqrt{3} \times \left(3^{-3}\right)^{-2} = 27^x$

Step 1: Simplify $\left(3^{-3}\right)^{-2}$

First, simplify the expression inside the parentheses using the power of a power rule $(a^m)^n = a^{m \times n}$: $\left(3^{-3}\right)^{-2} = 3^{-3 \times (-2)} = 3^6$

So the equation becomes: $\sqrt{3} \times 3^6 = 27^x$

Step 2: Convert $\sqrt{3}$ to exponential form

$\sqrt{3}$ can be written as $3^{1/2}$. Now, the equation is: $3^{1/2} \times 3^6 = 27^x$

Step 3: Combine the exponents on the left side

Use the rule $a^m \times a^n = a^{m+n}$ to combine the exponents on the left side: $3^{1/2 + 6} = 27^x$

Simplify the exponent on the left side: $3^{6.5} = 27^x$

Step 4: Express 27 as a power of 3

Since $27 = 3^3$, the equation becomes: $3^{6.5} = (3^3)^x$

Step 5: Simplify the right side

Use the power of a power rule again: $3^{6.5} = 3^{3x}$

Step 6: Equate the exponents

Since the bases are the same, you can equate the exponents: $6.5 = 3x$

Step 7: Solve for $x$

Divide both sides by 3: $x = \frac{6.5}{3} = \frac{13}{6} \approx 2.1667$

So, the solution is: $x = \frac{13}{6}$

Would you like any additional details or have any questions?

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Here are 8 related questions:

1. How do you solve equations involving square roots and exponents?
2. What are the properties of exponents that are essential for solving exponential equations?
3. How do you convert a radical expression into an exponential form?
4. What would happen if the exponent on 27 were negative?
5. Can you solve the equation $4 \times (2^{-2})^{-3} = 16^y$?
6. What steps would you take to solve $2^x = 32$ for $x$?
7. How would you solve the equation $9^x = 81$ using properties of exponents?
8. How does the equation change if you replace $\sqrt{3}$ with $\sqrt{2}$?

Tip: When working with exponential equations, expressing all terms with the same base simplifies the problem significantly.