Use the properties of exponents to simplify the following as much as possible. Assume the bases are positive.
(8a^3b^6)^(1/3) /
(16a^8b^−4)^(1/4)
Well, well, well, looks like we've got ourselves a fun exponent puzzle! Let's dive in and use some clown logic to simplify this expression.
First, let's tackle the numerator, which is (8a^3b^6)^(1/3). To simplify this, we need to raise each term within the parentheses to the power of 1/3.
The number 8 raised to the power of 1/3 is the same as taking the cube root of 8, which is 2. So, we're left with 2.
Now, for the variables inside, a^3 and b^6, we need to divide the exponents by 3. That means a^3 becomes a^(3/3) = a^1 = a, and b^6 becomes b^(6/3) = b^2.
So, the numerator is now 2ab^2.
Moving on to the denominator, which is (16a^8b^-4)^(1/4). We'll use similar clown logic to simplify this expression.
The number 16 raised to the power of 1/4 is the same as taking the fourth root of 16, which is 2. So, we're left with 2.
Now, let's handle the variables. a^8 raised to the power of 1/4 is the same as taking the fourth root of a^8, which simplifies to a^2. And b^-4 raised to the power of 1/4 is equal to taking the fourth root of b^-4, which simplifies to b^-1.
Putting it all together, the denominator is 2a^2b^-1.
Finally, we can simplify the entire expression by dividing the numerator by the denominator:
(2ab^2) / (2a^2b^-1)
Since we have a common factor of 2 in both the numerator and denominator, we can cancel it out:
(2/2)(ab^2)/(a^2b^-1) = ab^2/(a^2b^-1)
Now, to make things look even prettier, let's rearrange the terms a bit:
(ab^2)/(a^2b^-1) = (a/a^2)(b^2/b^-1)
When we divide variables with the same base, we subtract the exponents:
(a/a^2)(b^2/b^-1) = (1/a^(2-1))(b^(2-(-1)))
Simplifying further, we get:
(1/a)(b^3) = b^3/a
And viola! We've simplified our expression to b^3/a. Clown bot successfully strikes again!
To simplify the expression (8a^3b^6)^(1/3) / (16a^8b^−4)^(1/4), we can combine the exponents and simplify each base separately.
Step 1: Simplify the numerator.
Inside the parentheses, the base is 8, and the exponents are multiplied by 1/3. Applying the property of exponents (a^m)^n = a^(m*n), we get:
8^(1/3) * (a^3)^(1/3) * (b^6)^(1/3)
Since 8^(1/3) is equal to 2 (cubed root of 8), we have:
2 * a^(3 * 1/3) * b^(6 * 1/3)
2 * a^1 * b^2
2ab^2
Step 2: Simplify the denominator.
Inside the parentheses, the base is 16, and the exponents are multiplied by 1/4. Applying the property of exponents (a^m)^n = a^(m*n), we get:
16^(1/4) * (a^8)^(1/4) * (b^(-4))^(1/4)
Since 16^(1/4) is equal to 2 (fourth root of 16), we have:
2 * a^(8 * 1/4) * (b^(-4 * 1/4))
Simplifying the exponents, we get:
2 * a^2 * b^(-1)
Step 3: Divide the numerator by the denominator.
(2ab^2) / (2a^2 * b^(-1))
When dividing with the same base, we subtract the exponents:
2a^(1-2) * b^(2-(-1))
Simplifying, we get:
2a^(-1) * b^3
Finally, we can rewrite the expression as 2b^3 / a.
To simplify the given expression, we can use the properties of exponents.
First, let's simplify each term separately and then divide the results.
(8a^3b^6)^(1/3) can be simplified as follows:
The exponent 1/3 means taking the cube root of the expression.
Cube root of 8: The cube root of 8 is 2, as 2 x 2 x 2 = 8.
Cube root of a^3: Since the exponent inside the parentheses is already 3, the cube root cancels out with the exponent, resulting in just a.
Cube root of b^6: Similarly, the cube root of b^6 is b^2, as b^2 x b^2 x b^2 = b^6.
So, (8a^3b^6)^(1/3) simplifies to 2ab^2.
Now let's simplify the second term in the expression:
(16a^8b^-4)^(1/4):
Cube root of 16: The cube root of 16 is 2, as 2 x 2 x 2 = 16.
Cube root of a^8: Again, the cube root cancels out with the exponent, resulting in just a^2.
Cube root of b^-4: The cube root of b^-4 is b^-1 or 1/b^4.
So, (16a^8b^-4)^(1/4) simplifies to 2a^2/b.
Finally, to divide the two simplified terms, we'll divide each term individually:
(2ab^2) / (2a^2/b):
When dividing fractions, we multiply the first fraction by the reciprocal of the second fraction.
2ab^2 x (b/a^2):
Simplifying further, we have:
2ab^2b/a^2:
Now we can simplify the expression:
2 * a * b^2 * b / a^2:
This simplifies to:
2b^3/a
Therefore, the simplified expression is 2b^3/a.