2^(5/2)-2^(3/2)

Can someone show me in steps how to do exponents as fractions like in this problem ...thanks

Sure! To simplify expressions with exponents as fractions, you need to have a good understanding of the properties of exponents.

Let's break down the steps to simplify the expression 2^(5/2) - 2^(3/2:

Step 1: Understand the fractional exponent
First, we need to understand what the fractional exponents represent. Whenever you have an exponent in the form a^(b/c), it means taking the cth root of a raised to the power of b. In other words, it represents the b/c root of a.

Step 2: Convert the fractional exponent to a radical
To simplify the expression, we'll convert the fractional exponents to radicals. The numerator of the exponent will be the power of the base within the radical, and the denominator will be the index of the radical.

For 2^(5/2), the numerator is 5, and the denominator is 2. So, 2^(5/2) can be expressed as the square root of 2 raised to the power of 5.

Similarly, for 2^(3/2), the numerator is 3, and the denominator is 2. So, 2^(3/2) can be expressed as the square root of 2 raised to the power of 3.

Step 3: Simplify using the properties of exponents
Now that we have converted the fractional exponents to radicals, we can simplify the expression further.

2^(5/2) - 2^(3/2) becomes sqrt(2)^5 - sqrt(2)^3.

Remember, when subtracting expressions with the same base, we keep the base the same and subtract the exponents.

So, we have sqrt(2)^5 - sqrt(2)^3 = (sqrt(2) * sqrt(2) * sqrt(2) * sqrt(2) * sqrt(2)) - (sqrt(2) * sqrt(2) * sqrt(2)).

Step 4: Simplify the radicals
Now, we have multiples of the square root of 2. To simplify further, let's simplify the square roots:

(sqrt(2) * sqrt(2) * sqrt(2) * sqrt(2) * sqrt(2)) - (sqrt(2) * sqrt(2) * sqrt(2))
= (2 * 2 * sqrt(2)) - (2 * sqrt(2))
= (4 * sqrt(2)) - (2 * sqrt(2))
= 2 * sqrt(2)

Step 5: Final result
After simplifying, we have obtained the final result: 2 * sqrt(2).

So, the expression 2^(5/2) - 2^(3/2) simplifies to 2 * sqrt(2).

I hope this step-by-step explanation helps you understand how to simplify exponents with fractional exponents!