91Ó°ÊÓ

When you encounter division between exponential expressions with the same base, you can simplify it using the division law of exponents. This law states that when dividing two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. For example, if you have \(\frac{a^m}{a^n}\), you can write it as \[a^{m-n}\].
Let's apply this law to the given exercise: \(\frac{3^{5 / 2}}{3^{1 / 2}}\):
According to the division law, this becomes \[3^{(5 / 2) - (1 / 2)}\].
This simplification step is crucial for solving the problem efficiently.
Subtraction of exponents helps in reducing the complexity of the expression.

Simplifying exponents is another vital skill to master when dealing with exponential expressions. In our example \[(5 / 2) - (1 / 2)\], both exponents need to have the same denominator to simplify them properly.
Here, both fractions already have a common denominator of 2. By subtracting the numerators, we obtain: 5 - 1.
This simplifies to \[ \frac{4}{2} \], which can be further simplified to 2.
So, after subtraction, the expression \[3^{(5 / 2) - (1 / 2)}\] simplifies to \[3^{2}\].
This final step converts the original complex expression into a simpler form that's easier to understand and compute.

Exponential expressions, like the one in our problem, are mathematical expressions that involve exponents.They are written in the form \[a^n\], where 'a' is the base, and 'n' is the exponent.
Exponents indicate how many times the base is multiplied by itself. For example, \[3^2\] means 3 * 3.
Properly understanding and handling exponential expressions is crucial when simplifying and performing arithmetic operations.
In the exercise we solved, we transformed \[3^{5 / 2} / 3^{1 / 2}\] into a more manageable \[3^2\] by following exponent rules.
This emphasizes how the laws of exponents simplify complex expressions, making them easier to work with.
The final simplified form, \[3^2 = 9\], illustrates that working through these laws correctly leads to straightforward solutions.