Convert the following expression to exponential form: [tex]$\sqrt[11]{b^2}$[/tex]
Answer (2)
Recognize that a radical expression can be converted to exponential form.
Recall the rule n x m = x n m .
Apply the rule to the given expression 11 b 2 to get b 11 2 .
The exponential form of the given expression is b 11 2 .
Explanation
Understanding the Problem We are asked to convert the radical expression 11 b 2 into its equivalent exponential form.
Recalling the Rule Recall the relationship between radicals and exponents: n x m = x n m . This means the n -th root of x raised to the power of m is the same as x raised to the power of n m .
Applying the Rule to the Expression Applying this rule to our expression 11 b 2 , we identify x as b , m as 2 , and n as 11 . Therefore, we can rewrite the expression as b 11 2 .
Final Answer Thus, the exponential form of 11 b 2 is b 11 2 .
Examples
Imagine you are calculating the growth rate of a plant. If the growth can be modeled by the expression 5 x 3 , where x is some growth factor, converting this to exponential form x 5 3 makes it easier to analyze and compare with other growth models. Exponential forms are particularly useful in calculus and other advanced mathematical analyses where manipulating radicals can be cumbersome. This conversion simplifies the mathematical operations and provides a clearer understanding of the underlying relationships.
The expression 11 b 2 can be converted to exponential form using the rule that relates roots to exponents. Applying the rule, we find that 11 b 2 = b 11 2 . Thus, the exponential form is b 11 2 .
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