Select all the correct answers.
Which expressions are equivalent to the given expression?
[tex]$\sqrt{80}$[/tex]
[tex]$8 \sqrt{5}$[/tex]
[tex]$4 \sqrt{5}$[/tex]
[tex]$80^{\frac{1}{2}}$[/tex]
[tex]$4 \sqrt{10}$[/tex]
[tex]$160^{\frac{1}{2}}$[/tex]
Answer (1)
Simplify the given expression 80 by finding the largest perfect square factor.
Rewrite 80 as 16 × 5 = 4 5 .
Compare each expression to 4 5 to determine equivalence.
The equivalent expressions are 4 5 and 8 0 2 1 . 4 5 , 8 0 2 1
Explanation
Problem Analysis We are given the expression 80 and asked to identify equivalent expressions from the list: 8 5 , 4 5 , 8 0 2 1 , 4 10 , 16 0 2 1 .
Simplifying the Expression First, let's simplify 80 . We look for the largest perfect square that divides 80. We have 80 = 16 × 5 , where 16 is a perfect square ( 16 = 4 2 ). Therefore, we can write 80 = 16 × 5 = 16 × 5 = 4 5 .
Checking Each Expression Now, let's examine each of the given expressions:
8 5 : This is not equal to 4 5 .
4 5 : This is equal to 4 5 .
8 0 2 1 : This is the same as 80 , which is equal to 4 5 .
4 10 : This is not equal to 4 5 .
16 0 2 1 : We can simplify this as 160 = 16 × 10 = 16 × 10 = 4 10 . This is not equal to 4 5 .
Final Answer Therefore, the expressions equivalent to 80 are 4 5 and 8 0 2 1 .
Examples
Understanding equivalent expressions is crucial in various fields. For example, in physics, simplifying expressions can help in calculating the trajectory of a projectile. If the initial velocity is given by 80 m/s, it's easier to work with the simplified form 4 5 m/s for calculations. Similarly, in engineering, simplifying expressions can aid in designing structures or circuits more efficiently. Knowing that 80 is the same as 4 5 allows engineers to optimize their designs and calculations, ensuring accuracy and efficiency.
