Find the values of base \( x \) of the follow i. \( 3(14 x)=45 x \) ii. \( 314 x=103_{a} \)

To solve the equations given in different bases, we will first convert the numbers from their respective bases to base 10 and then solve for \( x \). ### i. \( 3(14_x) = 45_x \) 1. **Convert \( 14_x \) to base 10**: \[ 14_x = 1 \cdot x^1 + 4 \cdot x^0 = x + 4 \] 2. **Convert \( 45_x \) to base 10**: \[ 45_x = 4 \cdot x^1 + 5 \cdot x^0 = 4x + 5 \] 3. **Set up the equation**: \[ 3(x + 4) = 4x + 5 \] 4. **Distribute and simplify**: \[ 3x + 12 = 4x + 5 \] 5. **Rearrange the equation**: \[ 12 - 5 = 4x - 3x \] \[ 7 = x \] Thus, the value of base \( x \) is \( \boxed{7} \). ### ii. \( 314_x = 103_a \) 1. **Convert \( 314_x \) to base 10**: \[ 314_x = 3 \cdot x^2 + 1 \cdot x^1 + 4 \cdot x^0 = 3x^2 + x + 4 \] 2. **Convert \( 103_a \) to base 10**: \[ 103_a = 1 \cdot a^2 + 0 \cdot a^1 + 3 \cdot a^0 = a^2 + 3 \] 3. **Set up the equation**: \[ 3x^2 + x + 4 = a^2 + 3 \] 4. **Rearrange the equation**: \[ 3x^2 + x + 1 = a^2 \] This equation relates \( x \) and \( a \). To find specific values, we need additional information about \( a \) or \( x \). However, we can express \( a \) in terms of \( x \): \[ a = \sqrt{3x^2 + x + 1} \] Thus, the relationship between \( x \) and \( a \) is given by \( a = \sqrt{3x^2 + x + 1} \). Without additional constraints, we cannot find specific values for \( x \) and \( a \). If you have a specific value for \( a \) or additional constraints, please provide that information for further calculations.