Write the following with rational, positive exponents.
a. sqrt of 8 = 8^1/2
b. seventh root of 5^2 = 5^2/7
c. sqrt sqrt 6 = ???
d. fourth root of sqrt of 5^3 = ???
Can someone please check my work and help with "c and d"? Thanks.
c. sqrt sqrt 6 = 6^1/4
d. fourth root of sqrt of 5^3 = (5^3)^1/8
Are these correct?
Note: for "d" do I just leave it like that or can I further simplify?
Thank you.
a. and b. are correct
c.
Recall that the (a^b)^c=a^(b*c)
sqrt(sqrt(x))=(x^(1/2))^(1/2)=x^(1/4)
d.
using again (a^b)^c=a^(b*c)
(sqrt(5^3)^(1/4))
=((5^3)^(1/2))^(1/4)
=((5^(3/2))^(1/4)
=5^((3/2)*(1/4))
=5^(3/8)
a. Your work is correct. The square root of 8 can be written as 8 raised to the power of 1/2, which is equivalent to 8^1/2.
b. Your work is correct. The seventh root of 5^2 can be written as 5^2/7.
c. To simplify the expression "square root square root 6," we need to break it down step by step.
First, let's handle the inner square root. The square root of 6 can be written as 6^(1/2).
Now, we have "square root of 6" with an exponent of 1/2. To simplify this further, we can multiply the exponents. Since the square root is equivalent to raising to the power of 1/2, our expression becomes (6^(1/2))^(1/2).
Using the exponent rule, when we raise a power to another power, we multiply the exponents. So, (6^(1/2))^(1/2) simplifies to 6^(1/2 * 1/2).
Multiplying the exponents gives us 6^(1/4). Therefore, the expression "square root square root 6" is equivalent to 6 raised to the power of 1/4, which can be written as 6^1/4.
d. To simplify the expression "fourth root of square root of 5^3," we break it down step by step.
First, let's tackle the inner square root. The square root of 5^3 can be written as (5^3)^(1/2).
Using the exponent rule, we multiply the exponents, giving us 5^(3/2).
Now, we have "fourth root of 5^(3/2)." We can simplify this by multiplying the exponents again. The fourth root is equivalent to raising to the power of 1/4, so our expression becomes (5^(3/2))^(1/4).
Applying the exponent rule, we multiply the exponents to get 5^(3/8). Therefore, the expression "fourth root of square root of 5^3" is equivalent to 5 raised to the power of 3/8, which can be written as 5^3/8.