The following work models how to write the quotient $\frac{\sqrt[3]{4}}{\sqrt{2}}$ using rational exponents.
$\frac{\sqrt[3]{4}}{\sqrt{2}}=\frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{\left(2^2\right)^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}$
Note that, in order to use the quotient property of exponents, the bases must be the same.
Using the work shown on the left, what is $\frac{\sqrt[3]{4}}{\sqrt{2}}$ in simplest radical form?
Answer (2)
The expression 2 3 4 simplifies to 6 2 . This is obtained by rewriting the terms with rational exponents, applying the quotient rule, and converting back to radical form. The final answer is indeed 6 2 .
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Rewrite the expression using rational exponents: 2 3 4 = 2 2 1 4 3 1 .
Express the base as a power of 2: 2 2 1 4 3 1 = 2 2 1 ( 2 2 ) 3 1 = 2 2 1 2 3 2 .
Apply the quotient rule of exponents: 2 2 1 2 3 2 = 2 3 2 − 2 1 = 2 6 1 .
Convert back to radical form: 2 6 1 = 6 2 .
Explanation
Understanding the Problem We are given the expression 2 3 4 and we want to simplify it to its simplest radical form, using the steps provided.
The given steps are: 2 3 4 = 2 2 1 4 3 1 = 2 2 1 ( 2 2 ) 3 1 = 2 2 1 2 3 2
We need to continue from here.
Applying the Quotient Rule of Exponents We have the expression 2 2 1 2 3 2 . To simplify this, we use the quotient rule of exponents, which states that a n a m = a m − n . Therefore, 2 2 1 2 3 2 = 2 3 2 − 2 1 Now, we need to calculate the exponent 3 2 − 2 1 .
Calculating the Exponent To calculate 3 2 − 2 1 , we need to find a common denominator, which is 6. So, 3 2 − 2 1 = 6 4 − 6 3 = 6 1 Therefore, the expression simplifies to 2 6 1 .
Converting to Radical Form Now, we convert the rational exponent back to radical form. Recall that a n 1 = n a . So, 2 6 1 = 6 2 Thus, the simplest radical form is 6 2 .
Final Answer The simplest radical form of 2 3 4 is 6 2 .
Examples
Understanding how to simplify radical expressions with rational exponents is useful in various fields, such as engineering and physics, where complex calculations involving roots and powers are common. For instance, when dealing with wave equations or electrical circuits, simplifying expressions can make calculations more manageable and provide clearer insights into the underlying phenomena. This skill is also essential in computer graphics for rendering images and creating realistic visual effects, where efficient calculations are crucial for performance.
