In the realm of mathematics, exponents hold a special place. They allow us to represent and manipulate numbers concisely and powerfully. In this mathematical journey, we will explore a fascinating equation that involves exponents and algebraic expressions. The equation at hand is:
60= 60b=5 Our objective is to find the value of the expression:12^(1-a-b)/(2-2b)
This might seem complex at first glance, but we will uncover the solution to this intriguing equation through careful examination and mathematical techniques.
Exploring the Equations: 60^a = 3 and 60^b = 5
To understand this problem, we must first consider the two given equations:
60a=3
60b=5
These equations relate the exponents ‘a’ and ‘b’ to 3 and 5, respectively. Our task is to find a way to express ‘a’ and ‘b’ in terms of common bases so we can simplify and evaluate the desired expression.
60a=3
log60a=
alog60=
a=log3/log60
60b=5
log60b=log5
blog60=log5
b=log5/log60
this is the expression we need to find the value
12^(1-a-b)/(2-2b)
log3/log60- log5/log60/ 2-2* log5/log60
cancel the log 60 by log 60
log3-log60- log5/2(log60- log5)
For a complete solution, check the above video:
Conclusion
In this mathematical journey, we began with two exponential equations and an expression involving exponents. By simplifying and using exponents’ properties, we could relate the given equations to the expression we wanted to evaluate.This exercise showcases the power of algebraic manipulation and the elegant way mathematics allows us to solve complex problems. It serves as a reminder of mathematical principles’ beauty and versatility, enabling us to unravel even the most intricate of equations.
