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?

---

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.

Learn how to solve the equation sqrt(3) x (3^-3)^-2 = 27^x step-by-step with a detailed explanation. This problem involves exponents, square roots, and the power of a power rule. Suitable for high school students studying algebra and exponential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation