Given that [tex]$\sqrt{3}=1.732$[/tex] and without using a calculator, evaluate [tex]$\sqrt{27}(\sqrt{243}+\sqrt{16})$[/tex]
Answer (2)
After simplifying the square roots in the expression 27 ( 243 + 16 ) , we found it evaluates to 101.784 . The steps included simplifying each square root, substituting their values, and then calculating the final result using the value of 3 = 1.732 .
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Simplify 27 to 3 3 , 243 to 9 3 , and 16 to 4 .
Substitute the simplified values into the expression: 3 3 ( 9 3 + 4 ) .
Expand the expression: 27 × 3 + 12 3 = 81 + 12 3 .
Substitute the given value 3 = 1.732 and calculate the final value: 81 + 12 ( 1.732 ) = 101.784 .
The final answer is 101.784 .
Explanation
Understanding the Problem We are given the expression 27 ( 243 + 16 ) and the value 3 = 1.732 . Our goal is to evaluate the expression without using a calculator.
Simplifying the Square Roots First, we simplify the square roots individually.
27 = 9 × 3 = 9 × 3 = 3 3
243 = 81 × 3 = 81 × 3 = 9 3
16 = 4
Substituting the Simplified Values Now, we substitute these simplified values back into the original expression:
27 ( 243 + 16 ) = 3 3 ( 9 3 + 4 )
Expanding the Expression Next, we expand the expression:
3 3 ( 9 3 + 4 ) = 3 3 × 9 3 + 3 3 × 4 = 27 × 3 + 12 3 = 81 + 12 3
Substituting the Value of 3 Now, we substitute the given value of 3 = 1.732 :
81 + 12 3 = 81 + 12 ( 1.732 ) = 81 + 20.784 = 101.784
Final Answer Therefore, the value of the expression is 101.784 .
Examples
Understanding how to simplify and evaluate expressions with square roots is useful in many areas, such as physics and engineering. For example, when calculating the length of the diagonal of a square or the distance between two points, you often encounter square roots. Being able to manipulate these expressions without a calculator can help you make quick estimations and solve problems more efficiently. Imagine you are building a square frame and need to find the length of the diagonal to add support. If the sides of the square are 27 feet, you can use the above simplification techniques to find the diagonal length quickly.
