Complete the steps to simplify $\sqrt[5]{4} \cdot \sqrt{2}$. Rewrite using rational exponents.
$\checkmark \quad 2^{\frac{2}{6}} \cdot 2^{\frac{1}{2}}$
$4^5 \cdot 2^2$
$4^{\frac{1}{6}} \cdot 2^{\frac{1}{4}}$
COMPLETE
What is the least common denominator of the exponents?
$\square$
Answer (1)
Rewrite the expression using rational exponents: 5 4 ⋅ 2 = 2 5 2 ⋅ 2 2 1 .
Find the least common denominator (LCD) of the exponents 5 and 2, which is 10.
Rewrite the exponents with the common denominator: 5 2 = 10 4 and 2 1 = 10 5 .
The least common denominator of the exponents is 10 .
Explanation
Understanding the Problem We are asked to complete the steps to simplify 5 4 ⋅ 2 and find the least common denominator of the exponents after rewriting the expression using rational exponents.
Rewriting with Rational Exponents First, we need to rewrite the expression using rational exponents. We have 5 4 = 4 5 1 and 2 = 2 2 1 . Since 4 = 2 2 , we can rewrite 4 5 1 as ( 2 2 ) 5 1 = 2 5 2 . Thus, the expression becomes 2 5 2 ⋅ 2 2 1 .
Adding the Exponents To multiply terms with the same base, we add the exponents. So we need to add 5 2 and 2 1 . To do this, we need to find the least common denominator (LCD) of 5 and 2.
Finding the Least Common Denominator The least common denominator of 5 and 2 is 10. Therefore, we rewrite the fractions with the common denominator: 5 2 = 5 × 2 2 × 2 = 10 4 and 2 1 = 2 × 5 1 × 5 = 10 5 .
Simplifying the Expression Now we can add the exponents: 10 4 + 10 5 = 10 9 . So, the expression simplifies to 2 10 9 . The least common denominator of the exponents is 10.
Final Answer The least common denominator of the exponents is 10.
Examples
Understanding and simplifying expressions with rational exponents is crucial in various fields, such as physics and engineering, where dealing with roots and powers is common. For instance, when analyzing the behavior of waves or calculating the energy levels of quantum particles, you often encounter expressions involving fractional exponents. Simplifying these expressions allows for easier manipulation and interpretation of the results. Consider calculating the area of a fractal shape, where the dimensions might be expressed as fractional powers. Simplifying these powers helps in determining the fractal's scaling properties and understanding its behavior.
