Complete the steps to simplify $\sqrt[5]{4} \cdot \sqrt{2}$. Rewrite using rational exponents.
$2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}$
$4^5 \cdot 2^2$
$4^{\frac{1}{6}} \cdot 2^{\frac{1}{7}}$
Answer (2)
To simplify the expression 5 4 ⋅ 2 , we convert to rational exponents resulting in 2 5 2 ⋅ 2 2 1 . By adding the exponents, we find that the simplified expression is 2 10 9 .
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Rewrite the expression using rational exponents: 2 5 2 \t ⋅ 2 2 1 .
Add the exponents: 2 5 2 + 2 1 .
Find a common denominator and add the fractions: 5 2 + 2 1 = 10 4 + 10 5 = 10 9 .
The simplified expression is 2 10 9 .
Explanation
Problem Setup We are given the expression 5 4 \t ⋅ 2 and asked to simplify it using rational exponents. The first step provided is 2 5 2 \t ⋅ 2 2 1 . We need to continue from there.
Adding Exponents We have 2 5 2 \t ⋅ 2 2 1 . To multiply these terms, we need to add the exponents. So we have 2 5 2 + 2 1 .
Finding Common Denominator To add the fractions 5 2 and 2 1 , we need to find a common denominator, which is 10. So we rewrite the fractions as 10 4 and 10 5 . Thus, we have 2 10 4 + 10 5 .
Adding Fractions Now we add the fractions in the exponent: 10 4 + 10 5 = 10 9 . So the expression becomes 2 10 9 .
Final Result Therefore, the simplified expression is 2 10 9 .
Examples
Understanding and simplifying expressions with rational exponents is crucial in various fields, such as physics and engineering, where complex calculations involving roots and powers are common. For instance, when analyzing wave propagation or dealing with electrical circuits, simplifying expressions with rational exponents can make calculations more manageable and provide clearer insights into the behavior of the system. Consider calculating the impedance of an electrical circuit, where the impedance might involve square roots and other fractional powers. Simplifying these expressions allows engineers to efficiently determine the circuit's response to different frequencies.
