Write each expression with a rational exponent. Do not attempt to simplify.
(a) [tex]$\sqrt[11]{11 w}=[/tex]
(b) [tex]$\sqrt[8]{10 t}=\sqrt[8]{10 t}$ $\sigma^6[/tex]
(c) [tex]$2 \sqrt[7]{17 a}=[/tex]
Answer (2)
Rewrite radical expressions using rational exponents.
Apply the rule n x = x n 1 .
(a) 11 11 w = ( 11 w ) 11 1
(b) 8 10 t = ( 10 t ) 8 1
(c) 2 7 17 a = 2 ( 17 a ) 7 1
The expressions with rational exponents are: ( 11 w ) 11 1 , ( 10 t ) 8 1 , 2 ( 17 a ) 7 1
Explanation
Understanding Rational Exponents We need to rewrite the given expressions using rational exponents. Remember that a radical expression n x can be written as x n 1 , where n is the index of the radical.
Rewriting the First Expression (a) 11 11 w can be written as ( 11 w ) 11 1 .
Rewriting the Second Expression (b) 8 10 t can be written as ( 10 t ) 8 1 .
Rewriting the Third Expression (c) 2 7 17 a can be written as 2 ( 17 a ) 7 1 .
Final Answer Therefore, the expressions with rational exponents are: (a) ( 11 w ) 11 1 (b) ( 10 t ) 8 1 (c) 2 ( 17 a ) 7 1
Examples
Rational exponents are used in various fields, such as physics and engineering, to simplify complex equations and calculations. For example, in calculating the period of a pendulum, the formula involves a square root, which can be expressed as a rational exponent. Similarly, in electrical engineering, rational exponents are used to describe the relationship between voltage and current in certain circuit elements. Understanding rational exponents allows for easier manipulation and analysis of these equations.
The expressions rewritten with rational exponents are: (a) ( 11 w ) 11 1 , (b) ( 10 t ) 8 1 , and (c) 2 ( 17 a ) 7 1 . This conversion applies the rule that rewrites radical expressions as powers. Each exponent represents the reciprocal of the root's index.
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