Write each entire radical as a mixed radical in simplest form ) √243p10q
To simplify the given radical (√243p10q) in mixed radical form, we need to find the largest perfect square that divides 243.
The largest perfect square that divides 243 is 81 (9^2).
So, we can rewrite √243p10q as √(81 * 3)p10q.
Now we can break down the radical √(81 * 3) as √81 * √3, which simplifies to 9√3.
Therefore, the simplified radical (√243p10q) can be written as 9√3p10q.
To simplify the radical √243p^10q, we can start by finding perfect square factors of 243 and separate them from the remaining factors inside the radical.
Step 1: Simplify the perfect square factor
243 can be simplified as it has a perfect square factor:
243 = 81 * 3
So, we can rewrite the radical as:
√(81 * 3 * p^10 * q)
Step 2: Simplify the perfect square
Now, let's simplify the perfect square:
√81 = 9
The radical becomes:
9√(3 * p^10 * q)
Step 3: Simplify the remaining factors
No other perfect square factors can be simplified. However, we can rewrite p^10 as p^5 * p^5.
The radical is now:
9√(3 * p^5 * p^5 * q)
Step 4: Multiply the remaining factors outside the radical
To simplify further, we can simplify the factors outside the radical:
9 * p^5 = 9p^5
The radical expression simplifies to:
9p^5√(3q)
Therefore, the simplest form of the radical √243p^10q is 9p^5√(3q).
To write √243p^10q as a mixed radical in simplest form, let's break it down step by step:
Step 1: Simplify the number under the radical sign.
Start by finding the prime factorization of 243:
243 = 3 * 3 * 3 * 3 * 3 = 3^5
So, we can rewrite √243 as √(3^5).
Step 2: Simplify the variables under the radical sign.
Since p^10 is already simplified, we can leave it as it is.
So far, we have √(3^5 * p^10q).
Step 3: Separate out the perfect square factors from the remaining factors.
Among the factors inside the radical, one pair of 3's can be pulled out because it's a perfect square. We can also pull out p^5 because it's an even power.
Now, we have √(3^2 * 3 * p^5 * p^5 * q).
Simplifying further, we get 3 * p^5 * √(3q).
Therefore, √243p^10q can be written as 3p^5√(3q) in simplest form as a mixed radical.