Simplify. All variables represent nonnegative numbers.
1. 3^√8m^3
2. (a^2b^4)^1/23^√b^6
3. 3^√27x^6
If you meant: the cube root
∛(8m^3) then you would have
= 2m , not both 8 and m^3 are perfect cubes
2.
very confusing way of typing it, if you represent the square root correctly with an exponent of 1/3, why not represent the cube root with an exponent of 1/3
so we have ...
√(a^2 b^4)* ∛(b^6)
= (a b^2)(b^2)
= a b^4
3.
∛(27x^6)
=3x^2
Nice work there.
I remember in college when I read Burton's translation of the 1001 Arabian Nights. He said some of the stories were so convoluted that they required the services of a diviner, rather than an interpreter.
To simplify these expressions, we can use the properties of exponents.
1. To simplify 3^√8m^3, we can first rewrite 8 as 2^3. Then, we can rewrite the expression as (3^(1/3))(2^3)(m^3). Using the property that (a^m)(a^n) = a^(m+n), we can simplify this to 3^(1/3) * 2^(3) * m^(3). Since all variables represent nonnegative numbers, we can simplify further by evaluating the exponents: 3^(1/3) * 8 * m^3 = 8 * 3^(1/3) * m^3.
2. For (a^2b^4)^(1/2) / 3^√b^6, we simplify the numerator and leave the denominator as is for now. Since (a^m)^n = a^(mn), we have (a^2b^4)^(1/2) = a^(2 * 1/2) * b^(4 * 1/2) = ab^2. Now we can rewrite the expression as (ab^2) / 3^√b^6. Using the property (a^m) / (a^n) = a^(m-n), we can simplify this to (ab^2) * 3^(-√b^6).
3. To simplify 3^√27x^6, we can rewrite 27 as 3^3. Then, the expression becomes 3^(1/3) * (3^3)^(1/2) * x^6. Applying the property (a^m)^n = a^(mn), we simplify it to 3^(1/3) * 3^(3/2) * x^6. Using the property (a^m)(a^n) = a^(m+n), we can further simplify this expression to 3^(1/3 + 3/2) * x^6 = 3^(5/6) * x^6.
These are the simplified forms of the given expressions.