Convert the following expression to exponential form: [tex]$\sqrt[9]{a^4}$[/tex]
Answer (2)
Recognize that the expression is a radical.
Recall the conversion rule: n x m = x n m .
Apply the rule to rewrite the expression: 9 a 4 = a 9 4 .
State the final answer: a 9 4
Explanation
Understanding the Problem We are asked to convert the expression 9 a 4 to exponential form. This involves understanding the relationship between radicals and exponents.
Recalling the Rule Recall that a radical expression n x m can be written in exponential form as x n m . In other words, the index of the radical becomes the denominator of the fractional exponent, and the power of the radicand becomes the numerator.
Applying the Rule to the Expression Applying this rule to the given expression 9 a 4 , we identify n = 9 and m = 4 . Therefore, we can rewrite the expression as a 9 4 .
Final Answer Thus, the exponential form of 9 a 4 is a 9 4 .
Examples
Understanding how to convert between radical and exponential forms is useful in various areas of mathematics, such as simplifying expressions, solving equations, and calculus. For example, when dealing with derivatives or integrals involving radicals, it's often easier to convert the radicals to exponential form first. Consider the function f ( x ) = 3 x 2 . To find its derivative, it's helpful to rewrite it as f ( x ) = x 3 2 . Then, using the power rule, we find f ′ ( x ) = 3 2 x − 3 1 , which can be rewritten as f ′ ( x ) = 3 3 x 2 .
The expression 9 a 4 can be converted to exponential form as a 9 4 . This uses the radical to exponent conversion rule which states that n x m = x n m . Understanding this conversion is useful in various areas of mathematics.
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