to view answer.Radicals are merely another way of writing fractional exponents. When written this way, the denominator of the fraction in the exponent indicates the “root” that we are taking, and the numerator remains in the exponent. For instance:
$$
8^{\Large{\frac{2}{3}}} = \root{\Large 3} \of {8^ 2 }
$$
What is a root of a number? First, we have to identify which root we are talking about it. In the above example, we have a cube root. A cube root of a number is the value that we multiply together 3 times (or, cube) to get the number under the root. So, in the above example, the cube root of $$8^{\large 2} $$ is the number that we would multiply by itself 3 times to get $$8^{\large 2} $$. In this case, the answer is 4, since $$4^{\large 3} = 64$$. In general, the $$n^{\large {th}}$$ root of x, written $$\sqrt[{\Large n}]{\, x} $$ is the value that you would have to take to a power of n to get x. Here, n is called the index and x is called the radicand.
It is the accepted notation that a radical without an index number is called a square root, and is the same as if we were to put a 2 on the outside: $$\sqrt {\;} = \root{\Large 2} \of {\;} $$
Important Note: What number is squared to get 16? A correct answer is 4. However, another answer is correct, too. $$\left( { - 4} \right)^2 $$ is also equal to 16. Thus, both 4 and -4 are square roots of 16. It is important to remember this, especially when solving equations involving square roots. When we write $$\sqrt{16} $$, we refer to the principal square root of 16, which is +4. We are often most interested in the principal nth root of a number. This is the root that has the same sign as the number under the radical. Note that when we want to indicate both square roots of 16, we write $$ \pm \sqrt {16} $$. For the following examples, we will examine only principal square roots.
Radicals have many special properties that make it possible for us to simplify them. A simplified radical is easier to work with in a problem, and is also easier to understand as an answer. For instance if someone told you that a board was $$
\root{\Large 3} \of {8^{\large 2} }
$$ feet long, you would not have a clear picture in your mind about the length of the board. However, if we simplify this, we see that the board is 4 feet long – a much easier number to understand and to work with.
We will generalize the following properties. In the generalizations, we will let m and n be positive integers and x and y be real numbers. Note that none of these properties are new. Just try re-writing the properties as fractional exponents, and you will see that they directly follow from the Laws of Exponents.
Property 1:
$$
\root{\Large n} \of x \cdot \root{\Large n} \of y = \root{\Large n} \of {x \cdot y}
$$
This property says that we can move multiplication of radicals with the same index underneath the radical.
$$
\sqrt {12} \cdot \sqrt 3 = \sqrt {12 \cdot 3} = \sqrt {36} = 6
$$
(Note: 6 is the principal square root of 36 – remember that -6 is also a square root of 36, but $$\sqrt {36} $$ indicates only the principal square root.)
This also works in reverse. We may want to split up multiplication under a radical:
$$
\root {\Large 3} \of {16} = \root {\Large 3} \of 8 \cdot \root{\Large 3} \of 2 = 2 \cdot \root {\Large 3} \of 2
$$
Property 2:
$$
{\Large \frac{{\root n \of x \;}}{{\root n \of y\; }} }= \root {\Large n} \of {\large{\frac{x}{y}}}
$$
This property says that we can move division of radicals with the same index underneath the radical.
$$
{\Large\frac{{\sqrt {150} }}{{\sqrt 3 }}} = \sqrt {\large{\frac{{150}}{3}}} = \sqrt {50}
$$
We can simplify this further using Property 1:
$$
\sqrt {50} = \sqrt {25} \cdot \sqrt 2 = 5\sqrt 2
$$
Radicals with different indices: For Property 1 and Property 2, it is very important to remember that the radicals must have the same index in order to move the multiplication or division under the radical. For instance: $$
cannot be simplified. Why not? They have different exponents. They can be rewritten as: $$
Refer to Laws of Exponents for more explanation. |
Property 3:
$$
\root{\Large n} \of {x^ m } = \left( {\root{\large n} \of x } \right)^ m
$$
This property tells us that we can either take the radicand to a power before we take find the root, or we can find the root, and then take it to the power.
$$
\root {\Large 4} \of {16^ 3 } = \left( {\root{\Large 4} \of {16} } \right)^ 3 = \left( 2 \right)^ 3 = 8$$
In this example, it was much more convenient to find the 4th root of 16 and then cube it, since we know the 4th root of 16, but we probably do not know the 4th root of $$16^3 = 4096$$.
Property 4:
$$\sqrt[{\Large m}]{\sqrt[{\Large n}]{x}}=\sqrt[{\Large m \cdot n}]{x}$$
This property tells us that if we are taking a root of a root, we can multiply the indices together and get a new index.
$$
\root {\Large 3} \of {\root {} \of {128} } = \root {\Large{3 \cdot 2}} \of {128} = \root{\Large 6} \of {128}
$$
This can actually be simplified further, because $$128 = 2 \cdot 64$$.
So, $$\root{\Large 6} \of {128} = \root{\Large 6} \of {64} \cdot \root{\Large 6} \of 2 = 2 \cdot\root{\Large 6} \of 2 $$
Property 5:
$$
\left( {\root{\Large n} \of x } \right)^{\large n} = x
$$
This property just says that if we raise the nth root of a number to n, we will get that number. This is basically the definition of a root. After all, when are looking for an nth root of x, we are just looking for a number that, when raised to n, gives us x.
$$
\left( {\root {\Large {17}} \of {84} } \right)^{\large {17}} = 84
$$
Perfect squares, cubes, and recognizing powers A perfect square is a square that has an integer square root. For example, 25 is a perfect square because its square root is 5 (and -5, if we are interested in more than the principal square root). Similarly a perfect cube has an integer cube root. The more of these “perfect” numbers you are familiar with, the easier it will be for you to recognize numbers that can be simplified from under a radical. You should most certainly memorize the squares of the first 12 integers: It is also good to memorize as many cubes as you can, at least the first 5:
Some other useful powers to memorize are:
These tend to come up in problems, and it is useful to recognize them. |
Simplify:
$$
{\Large\frac{{\root 3 \of { - 72} }}{{\root 3 \of 9 }}}
$$
First, note that the indices are the same for both radicals. So, we take the radical of the division of the two numbers:
$$
{\Large\frac{{\root 3 \of { - 72} }}{{\root 3 \of 9 }}} = \root{\Large 3} \of {{\large\frac{{ - 72}}{9}}} = \root{\Large 3 }\of { - 8}
$$
Now, we have a negative number under our radical. If the index were even (for instance, if we were taking the square root instead of the cube root), this would be a problem because there is no real number that, when multiplied by itself an even number of times gives a negative result. However, since this is an odd index, we are fine:
$$
\root {\Large 3} \of { - 8}
$$
The principal, and in fact, the only cube root of -8 is -2. This is because $$ - 2 \cdot - 2 \cdot - 2 = \left( { - 2} \right)^3 = - 8$$.
Simplify:
$$
\root{\Large 3} \of {24a^4 }
$$
So, here, we have a variable under the radical. We can still deal with it the same way Let’s split up the multiplication under the radical to find some perfect cubes:
$$
\root {\Large 3} \of {24a^ 4 } = \root{\Large 3} \of {24 \cdot a^ 3 \cdot a} = \root {\Large 3} \of {8 \cdot 3 \cdot a^ 3 \cdot a}
$$
We notice that 8 and $$a^3 $$ are perfect cubes, so we will separate those parts of the multiplication:
$$
\root {\Large 3} \of {8 \cdot 3 \cdot a^3 \cdot a} = \root {\Large 3} \of 8 \cdot \root{\Large 3} \of {a^3 } \cdot \root {\Large 3} \of {3a}
$$
Finally, taking our cube roots, we get:
$$
\root {\Large 3} \of 8 \cdot \root{\Large 3} \of {a^3 } \cdot \root{\Large 3} \of {3a} = 2a \cdot \root {\Large 3} \of {3a}
$$
And, that is as simplified as we can make it.
Some practice problems to check your skills:
Simplify the following radical expressions (you can leave your answer as the principal nth root).
1. $$
\sqrt {\sqrt {32} }
$$
Think about it for a moment and then access this link to view answer.
2. $${\Large\frac{{\sqrt { - 7} }}{{\sqrt {14} }}}
$$
Think about it for a moment and then access this link to view answer.
3. $$
\sqrt {27x^7 a^6 } \cdot \sqrt {40x^2 a^2 }
$$
Think about it for a moment and then access this link to view answer.
4. $$
{\Large\left( {\frac{{\root 3 \of {4x^3 } }}{{\root 6 \of {a^3 } }}} \right)^2 }
$$
Think about it for a moment and then access this link to view answer.
5. $$
{\Large\frac{{\sqrt {2a^5 } }}{{\sqrt {8a^3 } }}}
$$
Think about it for a moment and then access this link to view answer.