Describe how to transform $\left(\sqrt[5]{x^7}\right)^3$ into an expression with a rational exponent. Make sure you respond with complete sentences.
Answer (2)
To rewrite ( 5 x 7 ) 3 with a rational exponent, first convert the fifth root to a rational exponent, resulting in x 5 7 . Substituting this back and using the power of a power rule gives us x 5 21 . Therefore, the final expression is x 5 21 .
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Rewrite the fifth root of x 7 using a rational exponent: 5 x 7 = x 5 7 .
Substitute this back into the original expression: ( x 5 7 ) 3 .
Use the power of a power rule to simplify the expression: ( x a ) b = x a ⋅ b .
Multiply the exponents: x 5 7 ⋅ 3 = x 5 21 . The final expression is x 5 21 .
Explanation
Understanding the Problem We are given the expression ( 5 x 7 ) 3 and asked to rewrite it with a rational exponent. This involves understanding fractional exponents and applying the power of a power rule.
Converting the Root to a Rational Exponent First, we need to express the fifth root of x 7 using a rational exponent. Recall that n x m = x n m . Therefore, 5 x 7 can be written as x 5 7 .
Substituting Back into the Expression Now, substitute this back into the original expression: ( x 5 7 ) 3 .
Applying the Power of a Power Rule Next, we use the power of a power rule, which states that ( x a ) b = x a ⋅ b . Applying this rule, we get ( x 5 7 ) 3 = x 5 7 ⋅ 3 .
Multiplying the Exponents Finally, we multiply the exponents: 5 7 ⋅ 3 = 5 7 ⋅ 3 = 5 21 . Therefore, the expression becomes x 5 21 .
Final Answer Thus, the expression ( 5 x 7 ) 3 can be transformed into x 5 21 , which has a rational exponent.
Examples
Rational exponents are used in various fields, such as physics and engineering, to simplify complex equations and calculations. For example, when dealing with wave equations or calculating growth rates, rational exponents provide a concise way to represent roots and powers, making the analysis more manageable. Understanding how to manipulate expressions with rational exponents is crucial for solving problems involving these concepts.
