7­5 Rational Exponents and Radicals index​
(noun) in deks Related Words:​
radical sign, radicand Main Idea:​
The index of a radical gives the degree of the root, so the cube root has an index of 3. If there is no index, the degree is 2. 4
Example:​
In the radical √16 , the index is 4, which means, “what number times itself four times is 16?” Essential Understanding​
You can rational exponents to represent radicals. In a radical expression, the number under the radical sign a is the radicand. The number n in the crook of the radical sign is the index . The index​
gives the degree of the root. For a cube root, the degree is 3. If there is no index, the degree is 2, which means square root. Recall what you know about square roots. Since 52 = 25 , you know 1
that √25 = 5 . You also know that 25 2 = 5 . Using the transitive 1
property of equality, you can conclude that √25 = 25 2 . Similarly, 1
√3 8 = 8 3 . You can simplify radical expressions by finding like factors, just as when simplifying powers with rational exponents. What is the value of each expression? 2
A.) √49
4
B.) √625
3
C.) √216
You can also write expressions that have rational exponents like 23
in radical form. 1
3
( )
8 = 8 = (8 ) = (√8) So, 8 = √8 = (√8) 2
3
2• 13
2
3
1 •2
3
8 =8
2
= 8
1
3
2
3
3
2
3
= √82 2
3
3
2
2
Write each expression in radical form. 2
1
D.) a 3
E.) 25x 2 1
F.) (25x) 2 1
G.) (98d) 2 2
3
H.) (24c) Write each expression in exponential form. I.) √
J.) (2c)4 √
3
(8x)2 4
K.) √
625y3 4
L.) √x3 M.) ​
Manufacturing​
A company that manufactures memory chips for 3
digital cameras uses the formula c = 120√n2 + 1300 to determine the cost c , in dollars, of producing n chips. How much will it cost to produce 250 chips?