Select all the correct answers.
Which expressions are equivalent to the given expression?
[tex]$\sqrt{40}$[/tex]
[tex]$160^{\frac{1}{2}}$[/tex]
[tex]$2 \sqrt{10}$[/tex]
[tex]$40^{\frac{1}{2}}$[/tex]
[tex]$5 \sqrt{8}$[/tex]
[tex]$4 \sqrt{10}$[/tex]
Answer (1)
Simplify the given expression: 40 = 2 10 .
Check 16 0 2 1 : 16 0 2 1 = 4 10 .
Check 2 10 : 2 10 = 2 10 .
Check 4 0 2 1 : 4 0 2 1 = 2 10 .
Check 5 8 : 5 8 = 10 2 .
Check 4 10 : 4 10 = 4 10 .
The equivalent expressions are: 2 10 , 4 0 2 1 .
Explanation
Understanding the Problem We are given the expression 40 and asked to identify equivalent expressions from the list: 16 0 2 1 , 2 10 , 4 0 2 1 , 5 8 , 4 10 .
Simplifying the Given Expression First, let's simplify the given expression 40 . We can write 40 as 4 × 10 , so 40 = 4 × 10 = 4 × 10 = 2 10 .
Checking Each Option Now, let's examine each of the options:
16 0 2 1 = 160 = 16 × 10 = 16 × 10 = 4 10 . This is not equal to 2 10 .
2 10 is already in simplified form and is equal to 2 10 .
4 0 2 1 = 40 = 4 × 10 = 4 × 10 = 2 10 . This is equal to 2 10 .
5 8 = 5 4 × 2 = 5 × 4 × 2 = 5 × 2 × 2 = 10 2 . This is not equal to 2 10 .
4 10 is not equal to 2 10 .
Final Answer Therefore, the expressions equivalent to 40 are 2 10 and 4 0 2 1 .
Examples
Understanding equivalent expressions is crucial in various fields, such as physics and engineering, where simplifying complex formulas is essential for calculations. For instance, when calculating the energy of a system, you might encounter an expression like 40 m , where m is the mass. Simplifying this to 2 10 m can make subsequent calculations easier and more intuitive. This skill is also useful in everyday situations, such as comparing prices or quantities in different units.
