Solving the Equation: 0.6^x * 0.6^3 = 0.6^(2x) / 0.6^5
In this article, we will explore the solution to the equation 0.6^x * 0.6^3 = 0.6^(2x) / 0.6^5. This equation involves exponentiation and algebraic manipulation, and we will break it down step by step.
Understanding the Equation
Before we dive into the solution, let's take a closer look at the equation:
0.6^x * 0.6^3 = 0.6^(2x) / 0.6^5
The equation consists of two expressions: the left-hand side (LHS) and the right-hand side (RHS). The LHS is the product of two terms: 0.6^x and 0.6^3. The RHS is a fraction, with the numerator being 0.6^(2x) and the denominator being 0.6^5.
Simplifying the Equation
To simplify the equation, we can start by using the property of exponents, which states that:
a^m * a^n = a^(m+n)
Using this property, we can rewrite the LHS as:
0.6^(x+3) = 0.6^(2x) / 0.6^5
Next, we can simplify the RHS by using another property of exponents, which states that:
a^m / a^n = a^(m-n)
Applying this property, we get:
0.6^(2x) / 0.6^5 = 0.6^(2x-5)
Now, we can equate the two expressions:
0.6^(x+3) = 0.6^(2x-5)
Solving for x
To solve for x, we can use the fact that the bases are the same (0.6) and the exponents are equal. Therefore, we can equate the exponents:
x + 3 = 2x - 5
Subtracting x from both sides gives:
3 = x - 5
Adding 5 to both sides yields:
8 = x
Therefore, the value of x is 8.
Conclusion
In conclusion, we have successfully solved the equation 0.6^x * 0.6^3 = 0.6^(2x) / 0.6^5, and found that the value of x is 8. This solution demonstrates the application of exponent properties and algebraic manipulation to solve complex equations.
