If all variables represent positive quantities, simplify the expression, using exact form
[tex]10 \sqrt{75 z^2}+2 z \sqrt{108}-\sqrt{27 z^2}=[/tex]
$\square$
NOTE: To input $\alpha \sqrt{b}$, type "a sqrt(b)". To input $\alpha \sqrt[3]{c}$, type "a root(b)(c)".
Answer (1)
Simplify 75 z 2 to 5 z 3 , so 10 75 z 2 = 50 z 3 .
Simplify 108 to 6 3 , so 2 z 108 = 12 z 3 .
Simplify 27 z 2 to 3 z 3 .
Combine the terms: 50 z 3 + 12 z 3 − 3 z 3 = 59 z 3 .
59 z 3
Explanation
Understanding the problem We are asked to simplify the expression 10 75 z 2 + 2 z 108 − 27 z 2 , where all variables represent positive quantities. This means that 0"> z > 0 . Our strategy will be to simplify each of the square root terms individually and then combine like terms.
Simplifying the first term Let's simplify the first term, 10 75 z 2 . We can rewrite 75 as 25 ⋅ 3 , so we have
10 75 z 2 = 10 25 ⋅ 3 ⋅ z 2 = 10 25 z 2 3 = 10 ⋅ 5 ⋅ z ⋅ 3 = 50 z 3 .
Simplifying the second term Now let's simplify the second term, 2 z 108 . We can rewrite 108 as 36 ⋅ 3 , so we have
2 z 108 = 2 z 36 ⋅ 3 = 2 z 36 3 = 2 z ⋅ 6 ⋅ 3 = 12 z 3 .
Simplifying the third term Next, we simplify the third term, 27 z 2 . We can rewrite 27 as 9 ⋅ 3 , so we have
27 z 2 = 9 ⋅ 3 ⋅ z 2 = 9 z 2 3 = 3 z 3 .
Combining the terms Now we combine the simplified terms:
10 75 z 2 + 2 z 108 − 27 z 2 = 50 z 3 + 12 z 3 − 3 z 3 = ( 50 + 12 − 3 ) z 3 = 59 z 3 .
Final Answer Therefore, the simplified expression is 59 z 3 .
Examples
Imagine you are calculating the total length of metal rods needed for a construction project. You have three types of rods with lengths expressed as 10 75 z 2 , 2 z 108 , and 27 z 2 , where z is a variable representing a base unit length. Simplifying the expression to 59 z 3 allows you to quickly determine the total length of rods needed for any value of z , making material estimation and ordering more efficient. This simplification transforms a complex calculation into a straightforward one, aiding in project planning and resource management.
