- Simplify the original expression: \(\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}\).
- Use the identity \(x^{\frac{1}{2}} = \sqrt{x}\) to verify \(40^{\frac{1}{2}} = \sqrt{40}\).
- Check the other options by squaring them or simplifying them into a standard radical form.
- The equivalent expressions are \(\boxed{40^{\frac{1}{2}}, 2\sqrt{10}}\).
Explanation
- Analysis and Simplification of the Expression
We are given the expression \(\sqrt{40}\) and need to determine which of the provided expressions are equivalent to it.
Recall the property of square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) for \(a, b \geq 0\).
Also, recall the definition of fractional exponents: \(x^{\frac{1}{2}} = \sqrt{x}\).
First, simplify the target expression \(\sqrt{40}\):
\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10}\).
Now, we evaluate each of the given options by simplifying or rewriting them.
- Evaluating the Options
- Option 1: \(4 \sqrt{10} = \sqrt{4^2 \times 10} = \sqrt{16 \times 10} = \sqrt{160}\). Since \(\sqrt{160}
eq \sqrt{40}\), this is not equivalent.
- Option 2: \(5 \sqrt{8} = \sqrt{5^2 \times 8} = \sqrt{25 \times 8} = \sqrt{200}\). Since \(\sqrt{200}
eq \sqrt{40}\), this is not equivalent.
- Option 3: \(40^{\frac{1}{2}} = \sqrt{40}\). This is equivalent to \(\sqrt{40}\).
- Option 4: \(2 \sqrt{10} = \sqrt{2^2 \times 10} = \sqrt{4 \times 10} = \sqrt{40}\). This is equivalent to \(\sqrt{40}\).
- Option 5: \(160^{\frac{1}{2}} = \sqrt{160}\). Since \(\sqrt{160}
eq \sqrt{40}\), this is not equivalent.
- Conclusion
Based on the evaluations, the expressions that are equivalent to \(\sqrt{40}\) are \(40^{\frac{1}{2}}\) and \(2 \sqrt{10}\). The final answer is \(\boxed{40^{\frac{1}{2}}, 2 \sqrt{10}}\).
Examples
Understanding how to simplify and identify equivalent radical expressions is essential in fields like engineering and physics, where complex formulas frequently involve square roots. For instance, if you are calculating the displacement of an object or the magnitude of a vector, you might reach a value such as \(\sqrt{40}\). Knowing that this is equal to \(2\sqrt{10}\) allows you to compare your result with other data or simplify the expression for easier calculation or interpretation in your final report. Mastering these properties helps ensure accuracy and efficiency in mathematical problem-solving.