Before you get started, take this readiness quiz.
Add: 7 15 + 5 12 . 7 15 + 5 12 .
If you missed this problem, review Example 1.28.
Simplify: ( 4 x 2 y 5 ) 3 . ( 4 x 2 y 5 ) 3 .
If you missed this problem, review Example 5.18.
Simplify: 5 −3 . 5 −3 .
If you missed this problem, review Example 5.14.
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that ( a m ) n = a m · n ( a m ) n = a m · n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that ( 8 p ) 3 = 8 . ( 8 p ) 3 = 8 . We will use the Power Property of Exponents to find the value of p.
( 8 p ) 3 = 8 Multiply the exponents on the left. 8 3 p = 8 Write the exponent 1 on the right. 8 3 p = 8 1 Since the bases are the same, the exponents must be equal. 3 p = 1 Solve for p . p = 1 3 ( 8 p ) 3 = 8 Multiply the exponents on the left. 8 3 p = 8 Write the exponent 1 on the right. 8 3 p = 8 1 Since the bases are the same, the exponents must be equal. 3 p = 1 Solve for p . p = 1 3
So ( 8 1 3 ) 3 = 8 . ( 8 1 3 ) 3 = 8 . But we know also ( 8 3 ) 3 = 8 . ( 8 3 ) 3 = 8 . Then it must be that 8 1 3 = 8 3 . 8 1 3 = 8 3 .
This same logic can be used for any positive integer exponent n to show that a 1 n = a n . a 1 n = a n .
If a n a n is a real number and n ≥ 2 , n ≥ 2 , then
a 1 n = a n a 1 n = a n
The denominator of the rational exponent is the index of the radical.
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
Write as a radical expression: ⓐ x 1 2 x 1 2 ⓑ y 1 3 y 1 3 ⓒ z 1 4 . z 1 4 .
AnswerWe want to write each expression in the form a n . a n .
| x 1 2 x 1 2 | |
| The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2. | x x |
| y 1 3 y 1 3 | |
| The denominator of the exponent is 3, so the index is 3. | y 3 y 3 |
| z 1 4 z 1 4 | |
| The denominator of the exponent is 4, so the index is 4. | z 4 z 4 |
Write as a radical expression: ⓐ t 1 2 t 1 2 ⓑ m 1 3 m 1 3 ⓒ r 1 4 . r 1 4 .
Write as a radial expression: ⓐ b 1 6 b 1 6 ⓑ z 1 5 z 1 5 ⓒ p 1 4 . p 1 4 .
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
Write with a rational exponent: ⓐ 5 y 5 y ⓑ 4 x 3 4 x 3 ⓒ 3 5 z 4 . 3 5 z 4 .
AnswerWe want to write each radical in the form a 1 n . a 1 n .
| 5 y 5 y | |
| No index is shown, so it is 2. The denominator of the exponent will be 2. | ( 5 y ) 1 2 ( 5 y ) 1 2 |
| Put parentheses around the entire expression 5 y . 5 y . |
| 4 x 3 4 x 3 | |
| The index is 3, so the denominator of the exponent is 3. Include parentheses ( 4 x ) . ( 4 x ) . | ( 4 x ) 1 3 ( 4 x ) 1 3 |
| 3 5 z 4 3 5 z 4 | |
| The index is 4, so the denominator of the exponent is 4. Put parentheses only around the 5 z 5 z since 3 is not under the radical sign. | 3 ( 5 z ) 1 4 3 ( 5 z ) 1 4 |
Write with a rational exponent: ⓐ 10 m 10 m ⓑ 3 n 5 3 n 5 ⓒ 3 6 y 4 . 3 6 y 4 .
Write with a rational exponent: ⓐ 3 k 7 3 k 7 ⓑ 5 j 4 5 j 4 ⓒ 8 2 a 3 . 8 2 a 3 .
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Simplify: ⓐ 25 1 2 25 1 2 ⓑ 64 1 3 64 1 3 ⓒ 256 1 4 . 256 1 4 .
Answer| 25 1 2 25 1 2 | |
| Rewrite as a square root. | 25 25 |
| Simplify. | 5 5 |
| 64 1 3 64 1 3 | |
| Rewrite as a cube root. | 64 3 64 3 |
| Recognize 64 is a perfect cube. | 4 3 3 4 3 3 |
| Simplify. | 4 4 |
| 256 1 4 256 1 4 | |
| Rewrite as a fourth root. | 256 4 256 4 |
| Recognize 256 is a perfect fourth power. | 4 4 4 4 4 4 |
| Simplify. | 4 4 |
Simplify: ⓐ 36 1 2 36 1 2 ⓑ 8 1 3 8 1 3 ⓒ 16 1 4 . 16 1 4 .
Simplify: ⓐ 100 1 2 100 1 2 ⓑ 27 1 3 27 1 3 ⓒ 81 1 4 . 81 1 4 .
Be careful of the placement of the negative signs in the next example. We will need to use the property a − n = 1 a n a − n = 1 a n in one case.
Simplify: ⓐ ( −16 ) 1 4 ( −16 ) 1 4 ⓑ − 16 1 4 − 16 1 4 ⓒ ( 16 ) − 1 4 . ( 16 ) − 1 4 .
Answer| ( −16 ) 1 4 ( −16 ) 1 4 | |
| Rewrite as a fourth root. | −16 4 −16 4 |
| ( −2 ) 4 4 ( −2 ) 4 4 | |
| Simplify. | No real solution. No real solution. |
| − 16 1 4 − 16 1 4 | |
| The exponent only applies to the 16. Rewrite as a fouth root. | − 16 4 − 16 4 |
| Rewrite 16 as 2 4 . 2 4 . | − 2 4 4 − 2 4 4 |
| Simplify. | −2 −2 |
| ( 16 ) − 1 4 ( 16 ) − 1 4 | |
| Rewrite using the property a − n = 1 a n . a − n = 1 a n . | 1 ( 16 ) 1 4 1 ( 16 ) 1 4 |
| Rewrite as a fourth root. | 1 16 4 1 16 4 |
| Rewrite 16 as 2 4 . 2 4 . | 1 2 4 4 1 2 4 4 |
| Simplify. | 1 2 1 2 |
Simplify: ⓐ ( −64 ) − 1 2 ( −64 ) − 1 2 ⓑ − 64 1 2 − 64 1 2 ⓒ ( 64 ) − 1 2 . ( 64 ) − 1 2 .
Simplify: ⓐ ( −256 ) 1 4 ( −256 ) 1 4 ⓑ − 256 1 4 − 256 1 4 ⓒ ( 256 ) − 1 4 . ( 256 ) − 1 4 .
We can look at a m n a m n in two ways. Remember the Power Property tells us to multiply the exponents and so ( a 1 n ) m ( a 1 n ) m and ( a m ) 1 n ( a m ) 1 n both equal a m n . a m n . If we write these expressions in radical form, we get
a m n = ( a 1 n ) m = ( a n ) m and a m n = ( a m ) 1 n = a m n a m n = ( a 1 n ) m = ( a n ) m and a m n = ( a m ) 1 n = a m n
This leads us to the following definition.
For any positive integers m and n,
a m n = ( a n ) m and a m n = a m n a m n = ( a n ) m and a m n = a m n
Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.
Write with a rational exponent: ⓐ y 3 y 3 ⓑ ( 2 x 3 ) 4 ( 2 x 3 ) 4 ⓒ ( 3 a 4 b ) 3 . ( 3 a 4 b ) 3 .
AnswerWe want to use a m n = a m n a m n = a m n to write each radical in the form a m n . a m n .
