We are asked to multiply the given radical expressions. Number one we have the
square root of 21 times the square root of 35. Looking at our notes, the nth root
of c times the nth root of d is equal to the nth root of c times d. So to multiply
radicals with the same index, we multiply the radicands.
In this case the square of 21 times the square root of 35 equals the square root of
21 times 35.
Because we need to simplify, we will not determine this product. We will determine
the prime factorization of 21 and 35. 21 is equal to 3 times 7 and 35 is equal to
5 x 7. Therefore this product has a prime factorization of 3 times 5 times 7
times 7.
Now that we have the prime factorization will identify the perfect square factors.
Every time we have two equal factors we have a perfect square factor. Therefore
seven times seven is a perfect square factor and this will simplify.
The square root of seven times seven or the square root of seven squared
simplifies the one factor of seven. Therefore this simplifies to seven
square root 15.
Number two, we have three square root 15 times negative 4 square root 75.
Looking back at our notes, a times the nth root of c times b times and root of d
equals a times b times the nth root of c times d, which means to find this product, we
multiply a and b outside the radical and then multiply c and d which are the
radicands under the radical.
So for number two, this product is equal to 3 times negative 4 times the square
root of 15 x 75. 3 times negative 4 is equal to negative 12
and we now determine the prime factorization of 15 and 75. 15 is equal to 3
3 times 5. 75 is equal to three times 25 and 25 is equal to 5 times 5.
The prime factorization of this product is 3 time 3 times 5 times 5 times 5. Now we
circle the perfect square factors of the radicand three times three is a perfect
square and so is five times five.
Simplifying we have negative 12 times the square root of three times three or
the square 3 squared is one factor of three, times the square root of 5 times 5
or the square of 5 squared is equal to one factor of five, and we still have
square root 5.
Negative 12 times 3 times 5 is equal to negative 180.
This simplifies to negative 180 square root of five. Now for number three, we
have the square root of 18 x to the third time's the square root of 10x
squared. This is equal to the square root of 18 times 10 times x raise to the power of 3 plus
2 which gives us x to the fifth. For the next step, we will determine the prime
factorization of 18 and 10. 18 is equal to two times nine. Nine is equal to 3 times 3. 10 is
equal to two times five.
For the prime factorization we have two factors of 2, two factors of 3 and a
factor of five. x to the fourth is a perfect square factor of X to the fifth.
To show this will write x to the fourth as x to the second times x to the second.
We still have x one more factor of x. Now we will circle the perfect square
factors, which are two times two, three times three and x squared times x squared. Simplifying
the square root of two x two is equal to one factor of two.
The square root of 3 times 3 is equal to one factor of three.
The square root of x squared times x squared is equal to one factor of x
squared. We're still left with the square root of 5x.
The final result is 6x squared times the square root of 5x. And because the
exponent on x is even no absolute value is required.
I hope you found this helpful.