Express each radical in exponential form. Simplify using the exponent laws when possible.
a) √k^2/3
b √ab^2
c 4√81x^8y^12
d 3√x √x
a) To express the radical √k^(2/3) in exponential form, we need to raise the base, k, to the power of the exponent of the radical, which is 1/2. However, before doing that, we simplify the exponent of k.
The exponent 2/3 can be expressed as (2/3) * 1 = 2/3. So, we rewrite k^(2/3) as k^(2/3) * 1.
Now, applying the exponent law, we know that x^(a * b) is equal to (x^a)^b. Therefore, we can rewrite k^(2/3) as (k^2)^(1/3).
Now, the radical √k^(2/3) can be rewritten as (k^2)^(1/3). Using the exponent law, we multiply the exponents within the parentheses to get k^(2/3 * 1/3) = k^(2/9).
Therefore, the expression √k^(2/3) in exponential form is k^(2/9).
b) To express the radical √ab^2 in exponential form, we need to raise the base, ab^2, to the power of the exponent of the radical, which is 1/2. However, before doing that, we simplify the exponent of b^2.
The exponent 2 can be expressed as 2 * 1/2 = 1. So, we rewrite b^2 as b^(2 * 1/2) = b^1.
Now, we can express the radical √ab^2 in exponential form as (ab^1)^(1/2). Using the exponent law, we multiply the exponents, giving us a^(1/2) * b^(1/2).
Therefore, the expression √ab^2 in exponential form is a^(1/2) * b^(1/2).
c) To express the radical 4√81x^8y^12 in exponential form, we need to raise the base, 81x^8y^12, to the power of the exponent of the radical, which is 1/4.
First, we simplify the exponent of 81. The exponent 1/4 can be expressed as 1 * 1/4 = 1/4. So, we rewrite 81 as 81^(1 * 1/4) = 81^(1/4).
Similarly, we simplify the exponents of x^8 and y^12. The exponents 8 and 12 can be expressed as 8 * 1/4 = 2 and 12 * 1/4 = 3, respectively. So, we rewrite x^8 as (x^2)^4 and y^12 as (y^3)^4.
Now, we can express the radical 4√81x^8y^12 in exponential form as (81^(1/4))(x^2)^4(y^3)^4. Expanding this expression using the exponent laws, we get 81^(1/4) * x^(2 * 4) * y^(3 * 4).
Simplifying further, we have 81^(1/4) * x^8 * y^12.
Therefore, the expression 4√81x^8y^12 in exponential form is 81^(1/4) * x^8 * y^12.
d) To express the radical 3√x √x in exponential form, we need to raise the base, x, to the power of the exponent of the radical, which is 1/3.
We have two radicals here, one 3√x and the other √x. Using the exponent laws, we can combine these radicals as (3√x * √x)^(1/3).
Using the exponent law, (3√x * √x)^(1/3) is equal to (x^(1/3) * x^(1/2))^(1/3).
Applying the exponent laws, (x^(1/3) * x^(1/2))^1/3 can be simplified to x^((1/3)*(1/3+1/2)).
Simplifying further, (x^(1/3) * x^(1/2))^1/3 becomes x^(1/3 * 1/3 + 1/3*1/2) = x^(1/9 + 1/6).
Combining the fractions, x^(1/9 + 1/6) becomes x^(2/18 + 3/18) = x^(5/18).
Therefore, the expression 3√x √x in exponential form is x^(5/18).
√k^2/3
=(k^(2/3) )^(1/2)
= k^(1/3)
b) is it √ab^2 , the way you typed it, or
√(ab^2) ?
first way would be a^(1/2) b^2
2nd way would be a^(1/2) b
c) and d)
same problem with interpretation.
is it:
4√81x^8y^12
or
4√(81x^8y^12)
etc