Multiply and simplify by factoring. Assume that all expressions under radicals represent nonnegative numbers.
∛(y^13 ) ∛(¡¼16y¡½^14 )
Am I supposed to add the exponents and multiply the little 3's?
To multiply and simplify the expression ∛(y^13) ∛(16y^14), you don't add the exponents and multiply the little 3's.
To simplify expressions with radicals, you need to use the rules of exponents and simplify each radical separately before multiplying.
Let's simplify each radical:
∛(y^13) = y^(13/3)
∛(16y^14) = (16y^14)^(1/3)
= 16^(1/3) * (y^14)^(1/3)
= 2 * y^(14/3)
Now, let's multiply the simplified radicals:
y^(13/3) * 2 * y^(14/3) = 2y^(13/3 + 14/3)
To add the exponents, we keep the same base (y) and add the exponents:
2y^(13/3 + 14/3) = 2y^(27/3)
= 2y^9
So, the simplified expression is 2y^9.
To multiply and simplify the expression ∛(y^13) * ∛(16y^14), you are correct that you need to multiply the little 3's and add the exponents.
First, let's simplify each cube root separately:
∛(y^13) = y^(13/3)
Next, let's simplify the second cube root:
∛(16y^14) = (16y^14)^(1/3) = (2^4 * y^14)^(1/3) = 2^(4/3) * y^(14/3)
Now, to multiply the two expressions together, we add the exponents:
y^(13/3) * (2^(4/3) * y^(14/3)) = 2^(4/3) * y^((13/3) + (14/3))
Simplifying the exponent:
= 2^(4/3) * y^(27/3)
= 2^(4/3) * y^9
Therefore, the simplified expression is 2^(4/3) * y^9, assuming all expressions under the radicals represent nonnegative numbers.