Help understanding surd concepts

Im having a little problem with this maths question and i dont know what im doing wrong, so please help me out that would be great (noob year 9er)

simplify 3√24p²q x 5√32pq^5 (sorry dont have small number signs for everything)

these are the steps i performed
15√768p³q^6

and then

240√3p³q^6

now the answer says 240pq³√3p
but i am not sure how to get to that answer? please help me


also 1 more question
(express as an entire surd)

3a²b³√5ab

answer is √45a^5b^7

but i did:
square everything in 3a²b³

so it becomes 9a^4b^9 and then multiply it back under the rational sign so its
√45a^3b^10 but that is wrong :(

THANKS IN ADVANCE !! :)

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1. For the first problem, you are doing OK.

      240√[3p³q⁶]            ← So far, so good ... Now, you know that √a² = a
      240√[3p²p(q³)²]            ← Rewrote the radicand to show p and q³ as squared
240pq³√[3p]            ← Used the rule √a² = a to pull out p and q³
240pq³√(3p)            ← ANSWER


2.    3a²b³√[5ab]            ← 3 = √9   So, put 9 inside the radicand
         a²b³√[9·5ab]            ← a² = √a⁴   So, put a⁴ inside the radicand
           b³√[45a⁴ab]            ← b³ = √b⁶   So, put b⁶ inside the radicand
               √[45a⁵b⁶b]
               √[45a⁵b⁷]            ← ANSWER


Hope that helps.
Have a good one!
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"Their" answer makes sense only if the original statement is:
3√(24p²q) × 5√(32pq^5)

Then you get 15√(768p³q^6), then 240√(3p³q^6), then finally, 240pq³√(3p)

Gotta watch those parentheses!

You made two errors. First, the square of 3a²b³ is 9a^4b^6.
Second, when you multiplied back under the *radical* sign, you turned a^4 back into a².