Given that 243√3 = 3^a, find the value of a.
(3 marks)^

To solve the equation \( 243\sqrt{3} = 3^a \), we will express both sides in terms of powers of 3.

1. **Express 243 as a power of 3**:
   
   \( 243 = 3^5 \) because:
   \[
   3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243
   \]

2. **Express \( \sqrt{3} \) as a power of 3**:
   
   The square root of 3 can be written as:
   \[
   \sqrt{3} = 3^{1/2}
   \]

3. **Combine the two expressions**:

   Now we substitute back into the original equation:
   \[
   243\sqrt{3} = 3^5 \times 3^{1/2}
   \]

4. **Use the laws of exponents to simplify**:

   When multiplying exponents with the same base, we add the exponents:
   \[
   3^5 \times 3^{1/2} = 3^{5 + 1/2}
   \]

5. **Calculate the exponent**:
   
   Convert the addition into a single fraction:
   \[
   5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}
   \]

   Therefore:
   \[
   3^{5 + 1/2} = 3^{11/2}
   \]

6. **Set equal to \( 3^a \) and solve for \( a \)**:
   
   Since \( 243\sqrt{3} = 3^a \) and we have expressed it as \( 3^{11/2} \), it follows that:
   \[
   a = \frac{11}{2}
   \]

Thus, the value of \( a \) is \(\boxed{\frac{11}{2}}\).

Would you like more details or have any questions?

Here are some related questions for further practice:

1. If \( 81 \times \sqrt{3} = 3^b \), what is the value of \( b \)?
2. Solve \( 27 \times \sqrt{27} = 3^c \) for \( c \).
3. Find the value of \( d \) if \( \frac{243}{\sqrt{3}} = 3^d \).
4. If \( 9\sqrt{3} = 3^e \), what is the value of \( e \)?
5. Evaluate \( 3^{2x} = 243 \) and find the value of \( x \).

**Tip:** When dealing with equations involving roots and exponents, try to express all numbers in terms of the same base, then use exponent rules to simplify.

Given that 243√3 = 3^a, find the value of a.

Discover how to solve the equation 243√3 = 3^a by breaking down exponents and radicals. Learn step-by-step how to express numbers in terms of powers of 3 and apply the laws of exponents to find the value of 'a'. Perfect for students in Grades 9-11 looking to strengthen their algebra skills.

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