Simplify: [tex]$\begin{aligned} \text { Simplify } & : 27_6-5^2 / 3+4^{3 / 4} \\ & =?\end{aligned}$[/tex]
Answer (2)
Convert 2 7 6 to base 10: 2 7 6 = 2 × 6 + 7 = 19 .
Evaluate 5 2 /3 = 25/3 ≈ 8.333 .
Evaluate 4 3/4 = 2 2 ≈ 2.828 .
Simplify the expression: 19 − 8.333 + 2.828 ≈ 13.495 . Therefore, the final answer is 19 − 3 25 + 2 2 .
Explanation
Understanding the Expression We are asked to simplify the expression 2 7 6 − 5 2 /3 + 4 3/4 . Let's break this down step by step.
Converting from Base 6 to Base 10 First, we need to convert 2 7 6 from base 6 to base 10. In base 6, 2 7 6 means ( 2 × 6 1 ) + ( 7 × 6 0 ) = ( 2 t im es 6 ) + ( 7 t im es 1 ) = 12 + 7 = 19 . So, 2 7 6 = 1 9 10 .
Evaluating the Exponent Next, we evaluate 5 2 , which is 5 t im es 5 = 25 .
Performing the Division Now, we divide the result by 3: 25/3 = 3 25 ≈ 8.333 .
Evaluating the Fractional Exponent Then, we evaluate 4 3/4 . This can be written as ( 4 1/4 ) 3 or ( 4 3 ) 1/4 . Let's calculate 4 3/4 = ( 2 2 ) 3/4 = 2 6/4 = 2 3/2 = 2 1 + 1/2 = 2 1 t im es 2 1/2 = 2 2 ≈ 2 t im es 1.414 = 2.828 .
Substituting the Values Now, we substitute these values back into the original expression: 19 − 3 25 + 2 2 = 19 − 8.333 + 2.828 .
Performing the Arithmetic Finally, we perform the subtraction and addition: 19 − 8.333 + 2.828 = 10.667 + 2.828 = 13.495 . Therefore, the simplified expression is approximately 13.495 .
Final Answer So, 2 7 6 − 5 2 /3 + 4 3/4 = 19 − 3 25 + 2 2 ≈ 13.495 .
Examples
Understanding different number bases and order of operations is crucial in computer science when dealing with binary, hexadecimal, or other base systems. Simplifying expressions like this helps in calculating memory addresses, data sizes, and other computational tasks. For example, converting a number from base 6 to base 10 can be useful when interfacing with systems that use different data representations. Evaluating exponents and fractions is fundamental in calculating algorithm complexities and resource allocation.
To simplify the expression 2 7 6 − 3 5 2 + 4 3/4 , convert the base 6 number to base 10, evaluate the powers and fractions, and perform the arithmetic. The final simplified result is approximately 13.495 .
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