I need to simplify the following rational exponent:
(64n^9)^2/3
I've done the following but don't think this is right:
= 3|64n^18
= 4n^2.62
I don't know how to type in the root symbol so I used | to mean root. Can someone please verify this and if incorrect please explain where I went wrong? Thank you very much...
(64n^9)^2/3
When you have an exponent of an exponent, you multiply the two. (9 * 2/3 = 6)
64^2/3 (n^6)
Cube root of 64 = 4 (4*4=16, 16*4 = 64)
4^2 = 16
(64n^9)^2/3 = 16n^6
To simplify the rational exponent (64n^9)^(2/3), we can use the exponent properties.
Recall that when you raise a power to another power, you multiply the exponents.
So we start by distributing the exponent of 2/3 to both the bases (64 and n^9):
(64n^9)^(2/3) = 64^(2/3) * (n^9)^(2/3)
Now let's simplify each term separately:
64^(2/3):
To simplify a number raised to a rational exponent, we take the denominator as the root and raise the numerator to that power.
In this case, the denominator is 3, so we take the cube root of 64, which is 4, and raise it to the power of 2: 4^2 = 16.
(n^9)^(2/3):
To simplify a variable raised to a rational exponent, we raise the variable itself to the power of the numerator and take the denominator as the root.
In this case, we raise n^9 to the power of 2: (n^9)^2 = n^(9*2) = n^18.
Putting it all together:
(64n^9)^(2/3) = 16 * n^18
So the simplified expression is 16n^18.
To simplify the given rational exponent (64n^9)^(2/3), you can follow these steps:
Step 1: Understand the rational exponent
The rational exponent 2/3 can be interpreted as "take the cube root of the entire expression and then square the result."
Step 2: Apply the rational exponent rule
By applying the rational exponent rule: (a^m)^n = a^(m*n), simplify the expression as follows:
(64n^9)^(2/3) = (64^(2/3)) * (n^9)^(2/3)
Step 3: Simplify the base
To simplify the base (64^(2/3)), recognize that 64 can be written as 4^3. Taking the cube root of 64 will result in 4.
(64^(2/3)) = (4^3)^(2/3)
Applying the exponent rule again will give you:
(4^3)^(2/3) = 4^((3*(2/3)))
Simplifying further:
4^((3*(2/3))) = 4^2 = 16
So, the base simplifies to 16.
Step 4: Simplify the exponent
To simplify the exponent (n^9)^(2/3), you need to multiply the exponents:
(n^9)^(2/3) = n^(9*(2/3))
Simplifying:
n^(9*(2/3)) = n^(18/3) = n^6
Step 5: Combine the simplified base and exponent
Now that you have simplified the base and exponent separately, you can combine them:
(64n^9)^(2/3) = (64^(2/3)) * (n^9)^(2/3) = 16 * n^6
Therefore, the simplified rational exponent of (64n^9)^(2/3) is 16n^6.
In summary:
(64n^9)^(2/3) simplifies to 16n^6.