Question

solve the given equation.
\\(left(6^{\frac{x}{3}}
ight)left(6^{\frac{x}{2}}
ight)=6^{4}\\)

\\(x = \square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Apply exponent product rule

When multiplying exponents with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$.
$$6^{\frac{x}{3} + \frac{x}{2}} = 6^4$$

Step2: Set exponents equal

Since the bases are equal, their exponents must be equal.
$$\frac{x}{3} + \frac{x}{2} = 4$$

Step3: Find common denominator

The least common denominator of 3 and 2 is 6.
$$\frac{2x}{6} + \frac{3x}{6} = 4$$

Step4: Combine like terms

Add the numerators over the common denominator.
$$\frac{5x}{6} = 4$$

Step5: Solve for x

Multiply both sides by $\frac{6}{5}$.
$$x = 4 \times \frac{6}{5} = \frac{24}{5}$$

Answer:

$\frac{24}{5}$