Question

$3^3 \times 3\sqrt{3} \div 3^{-\frac{5}{2}} = 3^{a+3}$

Then a = ?

• A. 3
• B. 4
• C. 1
• D. 2

Answer 1

Answer:

B. 4

Step 1: Convert all terms on the Left-Hand Side (LHS) into powers of 3

• $3^3$ stays the same.

• $3\sqrt{3} = 3^1 \times 3^{\frac{1}{2}} = 3^{1 + \frac{1}{2}} = 3^{\frac{3}{2}}$

• The divisor is $3^{-\frac{5}{2}}$

Now rewrite the LHS:
$\text{LHS} = 3^3 \times 3^{\frac{3}{2}} \div 3^{-\frac{5}{2}}$

Step 2: Apply the laws of exponents

• Multiply terms by adding their powers ($x^m \times x^n = x^{m+n}$)

• Divide terms by subtracting their powers ($x^m \div x^n = x^{m-n}$)

$\text{LHS} = 3^{\left(3 + \frac{3}{2} - \left(-\frac{5}{2}\right)\right)}$
$\text{LHS} = 3^{\left(3 + \frac{3}{2} + \frac{5}{2}\right)}$

Simplify the fraction part:
$\frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4$

Add it back to the whole number power:
$\text{LHS} = 3^{3 + 4} = 3^7$

Step 3: Equate both sides to find $a$
$3^7 = 3^{a+3}$

Since the bases are identical, their powers must be equal:
$7 = a + 3$
$a = 7 - 3$
$a = 4$