Simplify each expression
(-216)^1/3
My answer is 1/6. Am I correct.
(x^4y)^1/3 (xy^4)^2/3
I do not know how to simplify this.
(-216)^1/3 = - 1/6
(how can (1/6)(1/6)(1/6) end up negative ?
(x^4y)^1/3 (xy^4)^2/3
= (x^(13))(y^(1/3))(x^(2/3))(y^(8/3)
= x^2 y^3
1/6 is almost correct, except that you probably notice the original expression is negative. Give it one more try.
Use the rules of exponentiation:
(ab)^(m/n)=(a^m * b^m)^(1/n)
and the inverse:
(a^m)(1/n)
=(a^(m/n))
So
(x^4y)^2/3 (xy^4)^1/3
=( (x^4y)^2 * (xy^4) )^(1/3)
=( x^8*y^2 * x * y*4 )^(1/3)
=(x^9 * y^6)^(1/3)
=x^3 * y^2
(Note that the above example is not the same as the given question).
Post your answer for a check if you wish.
To simplify the expression (-216)^(1/3), we can evaluate the cube root of -216.
The cube root of -216 is -6, because (-6) x (-6) x (-6) = -216.
So, (-216)^(1/3) simplifies to -6.
Regarding the expression (x^4y)^(1/3) (xy^4)^(2/3), we can simplify it step by step.
First, let's simplify (x^4y)^(1/3):
Using the rule (ab)^n = a^n * b^n, we can break it down as follows:
(x^4y)^(1/3) = (x^4)^(1/3) * y^(1/3)
The cube root of x^4 is x^(4/3), and the cube root of y is y^(1/3).
Therefore, (x^4y)^(1/3) simplifies to x^(4/3) * y^(1/3).
Next, let's simplify (xy^4)^(2/3):
Using the same rule, we have:
(xy^4)^(2/3) = (x^1y^4)^(2/3)
Now, apply the rule (a^m)^n = a^(m*n):
(xy^4)^(2/3) = x^(1*2/3) * y^(4*2/3)
Simplifying further:
(xy^4)^(2/3) = x^(2/3) * y^(8/3)
Therefore, the simplified expression is x^(4/3) * y^(1/3) * x^(2/3) * y^(8/3).
Please note that if there are any additional simplifications, such as combining like terms, they would depend on the context of the problem.
To simplify the expression (-216)^(1/3), you can use the property of exponents which states that (a^m)^n = a^(m*n). In this case, the exponent inside the parentheses can be multiplied with the exponent outside.
So, (-216)^(1/3) can be written as (-216)^(1 * 1/3). Multiplying the exponents gives us (-216)^(1/3), which simplifies to -216^(1/3).
To evaluate -216^(1/3), you can factor out the cube root of 216, which is the number that, when raised to the third power, equals 216.
The cube root of 216 is 6 because 6^3 = 216.
Next, we can rewrite -216 as (-1) * 216 to factor out the cube root of 216: (-1) * (6^3).
Applying the property of exponents again, (-1 * 6^3)^(1/3) simplifies to (-1)^(1/3) * 6^(3/3).
Since (-1) raised to any odd power gives -1, (-1)^(1/3) equals -1.
Therefore, the simplified form of (-216)^(1/3) is -1 * 6^(3/3) = -6.
As for the expression (x^4y)^(1/3) * (xy^4)^(2/3), we can use the property of exponents that states (a * b)^n = a^n * b^n.
Hence, we can write the expression as x^(4*(1/3)) * y^(1*(1/3)) * x^(1*(2/3)) * y^(4*(2/3)).
Applying the exponent multiplication rule, we get x^(4/3) * y^(1/3) * x^(2/3) * y^(8/3).
Now, we can simplify further by adding the exponents with the same base:
x^(4/3) * x^(2/3) = x^((4/3) + (2/3)) = x^(6/3) = x^2.
Similarly, y^(1/3) * y^(8/3) = y^((1/3) + (8/3)) = y^(9/3) = y^3.
Hence, the simplified form of (x^4y)^(1/3) * (xy^4)^(2/3) is x^2 * y^3.