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 can use the property of radicals which states that ∛(a) * ∛(b) = ∛(a * b).
Step 1: Combine the expressions under the radicals by multiplying them.
∛(y^13) * ∛(16y^14)
= ∛(y^13 * 16y^14)
Step 2: Simplify the expression by adding the exponents.
= ∛(16y^27)
Step 3: Rewrite the expression under the radical in its prime factorization form.
= ∛(2^4 * y^3 * y^3 * y^3 * y^3 * y^3 * y^3 * y^3 * y^3 * y^3 * y)
Step 4: Take out groups of three similar factors from the radical.
= ∛(2^4 * y^9 * y^9 * y^9 * y)
Step 5: Simplify the expression inside the radical.
= ∛(16 * y^9 * y^9 * y^9 * y)
Step 6: Multiply the numbers outside the radical.
= 2 * y^3 * y^3 * y^3 * ∛(y)
Step 7: Combine the variables with the same base by adding the exponents.
= 2y^3 * y^3 * y^3 * y * ∛(y)
Step 8: Simplify the expression by adding the exponents.
= 2y^(3+3+3+1) * ∛(y)
Step 9: Final simplified expression.
= 2y^10 * ∛(y)
To simplify the expression, you can use the rules of radical multiplication and simplification.
In this case, you need to multiply the two cube roots ∛(y^13) and ∛(¡¼16y¡½^14). Since both cube roots have the same index (3), you can multiply the radicands (the expressions inside the cube roots) directly.
Multiplying the radicands gives us: (y^13) * (¡¼16y¡½^14)
To simplify this expression further, you can add the exponents of the variable y:
(y^13) * (¡¼16y¡½^14) = y^(13 + 14) * (¡¼16)
Simplifying the exponent of y, we have:
y^(27) * (¡¼16)
Finally, by factoring the cube root of y^27, we get:
∛(y^13) * ∛(¡¼16y¡½^14) = y^(27/3) * ∛(¡¼16) = y^9 * ∛(¡¼16)
So, the simplified expression is y^9 * ∛(¡¼16). It is not necessary to add the exponents or multiply the "little 3's" when multiplying cube roots.