(x^2+8xy+16y^2)^(1/3)*(x+4y)^(1/3)
Note that
(x^2+8xy+16y^2) = (x+4y)^2
so, you have
(x+4y)^(2/3) * (x+4y)^(1/3) = x+4y
To simplify the expression `(x^2+8xy+16y^2)^(1/3)*(x+4y)^(1/3)`, you can break it down into several steps.
Step 1: Factor the expression inside the first cube root.
The expression inside the first cube root `(x^2+8xy+16y^2)` can be factored as a perfect square trinomial: `(x+4y)^2`.
Step 2: Apply the product rule for exponents.
Using the product rule for exponents, we can write the expression as `(x+4y)^(2/3)*(x+4y)^(1/3)*(x+4y)^(1/3)`.
Step 3: Simplify the expression using exponent rules.
When we multiply two expressions with the same base, we add their exponents. Applying this rule, we get `(x+4y)^(2/3+1/3+1/3)`.
Step 4: Add the exponents.
Adding the exponents, we have `(x+4y)^(2/3+3/3)`.
Step 5: Simplify the exponents.
The sum of the exponents is `2/3 + 3/3 = 5/3`.
Step 6: Final result.
Therefore, the simplified expression is `(x+4y)^(5/3)`.
To simplify the given expression, (x^2+8xy+16y^2)^(1/3)*(x+4y)^(1/3), we can use the property of exponents that states (a^m * b^n)^(1/k) = (a^(m/k) * b^(n/k)).
First, let's break down each term inside the parentheses:
x^2+8xy+16y^2 can be factored as (x+4y)^2.
x+4y does not factor any further.
Now, let's substitute these expressions back into the given expression:
((x+4y)^2)^(1/3) * (x+4y)^(1/3)
Next, we can use the property of exponents mentioned above:
= (x+4y)^(2/3) * (x+4y)^(1/3)
Now, we can combine the two terms with the same base by adding their exponents:
= (x+4y)^((2/3)+(1/3))
= (x+4y)^(3/3)
Since the exponent 3/3 is equal to 1, we have:
= (x+4y)^1
Therefore, the simplified expression is x + 4y.