So far, we know the following:
To find the square root of a number, we need to find another that multiplies by itself times to give the original number. For example,
To find the cube root of a number, we need to find another that multiplies by itself times to give the original number. For example,
We can extend this to any root. In general, to find the th root of a number, we needed to find another number that multiplies by itself times to give the original number. We express the th root as the radical For instance,
Given a radical we can write an equivalent expression using the fractional exponent To demonstrate, let's write an equivalent expression for
Comparing with we get and Therefore,
The reverse is also true: the fractional exponent can be written as the radical
Note: Remember that the square root of a number corresponds to the case of So, for example, we have
Write using radical notation.
The fractional exponent can be written as the radical
Comparing with we get and Therefore,
Write $6^{1/4}$ using radical notation.
|
a
|
$\sqrt[4]{\dfrac 16}$ |
|
b
|
$\sqrt{6^4}$ |
|
c
|
$\sqrt[4]{6}$ |
|
d
|
$\sqrt[6]{4}$ |
|
e
|
$\sqrt[6]{\dfrac 14}$ |
Write $5^{1/5}$ using radical notation.
|
a
|
$5^5$ |
|
b
|
$\sqrt[5]{5}$ |
|
c
|
$1$ |
|
d
|
$\dfrac{1}{5^5}$ |
|
e
|
$\sqrt[5]{\dfrac{1}{5}}$ |
Express using a fractional exponent.
The radical can be written using a fractional exponent as
Comparing with we get and Therefore,
$\sqrt[7]{2} = $
|
a
|
$1^{2/7}$ |
|
b
|
$7^{1/2}$ |
|
c
|
$2^{1/7}$ |
|
d
|
$7^{2/1}$ |
|
e
|
$2^{7/1}$ |
$\sqrt{10} = $
|
a
|
$2^{{10}/1}$ |
|
b
|
$5$ |
|
c
|
$10^{2/1}$ |
|
d
|
$10^{1/2}$ |
|
e
|
$2^{1/{10}}$ |
Write using radical notation.
First, using the power law for exponents, we can write our expression as
The fractional exponent can be written as the radical
Comparing with our expression above, we get and Therefore,
$4^{7/6} = $
|
a
|
$\sqrt[4]{7^6}$ |
|
b
|
$\sqrt[7]{4^6}$ |
|
c
|
$\sqrt[6]{7^4}$ |
|
d
|
$\sqrt[7]{6^4}$ |
|
e
|
$\sqrt[6]{4^7}$ |
$2^{4/3} = $
|
a
|
$\sqrt[3]{16}$ |
|
b
|
$\sqrt[4]{9}$ |
|
c
|
$\sqrt{27}$ |
|
d
|
$\sqrt[3]{48}$ |
|
e
|
$\sqrt[4]{8}$ |
Express using a fractional exponent.
The radical can be written using a fractional exponent as
Comparing with we get and Therefore,
Finally, using the power law for exponents, we have
$\sqrt[4]{3^5} = $
|
a
|
$3^{5/4}$ |
|
b
|
$5^{4/3}$ |
|
c
|
$3^{4/5}$ |
|
d
|
$4^{3/5}$ |
|
e
|
$5^{3/4}$ |
$\sqrt{4^3} = $
|
a
|
$2^{3/4}$ |
|
b
|
$4^{3/2}$ |
|
c
|
$2^{4/3}$ |
|
d
|
$3^{2/4}$ |
|
e
|
$4^{2/3}$ |