Final answer:
To solve (2(x)/5)(2(x)/2) = 2^14, add the exponents of the base 2 after expressing them with a common denominator, which leads to the solution x = 20 after simplifying the equation.
Step-by-step explanation:
To solve the equation (2(x)/5)(2(x)/2) = 214, we must recognize that when we have two exponential expressions with the same base being multiplied, we can add their exponents. This is similar to understanding that (53)4 can be expressed as 53×4, which results in 512. Thus, we effectively just multiply the two exponents to obtain the combined exponent.
Applying this knowledge:
- Multiply the exponents of the two expressions with the base of 2, which are x/5 and x/2.
- Add these exponents to obtain the new exponent for the base of 2.
- Set this new expression equal to 214 and solve for x.
Performing the calculation:
- x/5 + x/2 = 14
- To combine fractions, find a common denominator, which in this case is 10.
- (2×x)/10 + (5×x)/10 = 14
- Combine the terms to get (2x + 5x)/10 = 14
- This simplifies to 7x/10 = 14
- Multiply both sides by 10 and divide by 7 to isolate x, yielding x = 20
Thus, the solution is x = 20.