Rewrite with rational exponents.
$\sqrt[3]{x y^2 z}$
Answer (2)
Rewrite the cube root as a power of 3 1 : ( 3 x y 2 z ) = ( x y 2 z ) 3 1 .
Apply the power to each term inside the parenthesis: ( x y 2 z ) 3 1 = x 3 1 ( y 2 ) 3 1 z 3 1 .
Simplify the exponents: x 3 1 y 3 2 z 3 1 .
The expression 3 x y 2 z rewritten with rational exponents is x 3 1 y 3 2 z 3 1 .
Explanation
Understanding the Problem We are asked to rewrite the expression 3 x y 2 z using rational exponents. This involves understanding how radicals and exponents are related.
Converting Radical to Rational Exponent Recall that a radical can be expressed as a rational exponent. Specifically, n a = a n 1 . In our case, we have a cube root, so n = 3 . Therefore, we can rewrite the given expression as: ( 3 x y 2 z ) = ( x y 2 z ) 3 1 .
Applying the Exponent to Each Term Now, we apply the power of 3 1 to each term inside the parentheses. Remember that when you raise a product to a power, you raise each factor to that power: ( a b ) n = a n b n . Applying this rule, we get: ( x y 2 z ) 3 1 = x 3 1 ( y 2 ) 3 1 z 3 1 .
Simplifying the Exponents Next, we simplify the exponent of y . Recall that when you raise a power to a power, you multiply the exponents: ( a m ) n = a m × n . So, we have: ( y 2 ) 3 1 = y 2 × 3 1 = y 3 2 . Therefore, the expression becomes: x 3 1 y 3 2 z 3 1 .
Final Answer Thus, the expression 3 x y 2 z rewritten with rational exponents is x 3 1 y 3 2 z 3 1 .
Examples
Rational exponents are useful in various fields, such as physics and engineering, for simplifying complex equations and calculations. For example, when dealing with wave equations or calculating stress and strain in materials, rational exponents can help in expressing relationships between variables in a more manageable form. Understanding how to manipulate and simplify expressions with rational exponents is a fundamental skill in these areas.
The expression 3 x y 2 z can be rewritten using rational exponents as x 3 1 y 3 2 z 3 1 . This conversion involves expressing the cube root as a power of 3 1 and applying this exponent to each variable. The final simplified expression is x 3 1 y 3 2 z 3 1 .
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