Multiplying sq rts
sqrt18a^7b times sqrt27a^8b^6
Jake 1214-
18 = 9 * 2 then sqrt 18 = 3sqrt2
sqrt a^7 = a^3 * sqrt a
27 = 9 * 3 then sqrt 27 = 3 sqrt3
sqrt a^8 = a^4
sqrt b^6 = b^3
Now just multiply the like-terms together.
3sqrt6 and sqrt2a^4?
.. 3sqrt6 and sqrt2a^4?
So you simplified the problem first as I said above and you get:
3(a^3)sqrt(2a) * 3(a^4)(b^3)sqrt(3)
and then you multiply those 2 things together and you get:
9(a^7)(b^3)sqrt(6a)
To multiply the square roots sqrt(18a^7b) and sqrt(27a^8b^6), you can follow these steps:
1. Simplify the square roots individually:
- For sqrt(18), break it down into its prime factors: 18 = 9 * 2 = (3^2) * 2 = 3sqrt(2).
- For sqrt(a^7), take out the highest even exponent possible from inside the square root: a^7 = a^6 * a = (a^3)^2 * a = a^3 * sqrt(a).
- For sqrt(b), since there is no higher even exponent available, it will remain as sqrt(b).
2. Simplify sqrt(27):
- Break 27 down into its prime factors: 27 = 9 * 3 = (3^2) * 3 = 3sqrt(3).
- For sqrt(a^8), take out the highest even exponent possible from inside the square root: a^8 = (a^4)^2 = a^4.
- For sqrt(b^6), take out the highest even exponent possible from inside the square root: b^6 = (b^3)^2 = b^3.
3. Now, multiply the simplified terms together:
sqrt(18a^7b) * sqrt(27a^8b^6) equals (3sqrt(2) * 3sqrt(3)) * (a^3 * a^4) * (sqrt(a) * b^3).
- Multiply the coefficients outside the square root: 3 * 3 = 9.
- Multiply the like variables inside the square root: sqrt(2) * sqrt(3) = sqrt(6).
- Multiply the powers of 'a': a^3 * a^4 = a^(3+4) = a^7.
- Multiply the powers of 'b': b^3.
So, the final answer is 9sqrt(6)a^7b^3.