Find the value of the expression: [tex]$0.2^6 \cdot 50^7$[/tex]
Answer (2)
Rewrite 0.2 as 5 1 and 50 as 5 × 10 .
Substitute these values into the expression: ( 5 1 ) 6 ⋅ ( 5 ⋅ 10 ) 7 .
Apply exponent rules to simplify: 5 6 1 ⋅ 5 7 ⋅ 1 0 7 = 5 ⋅ 1 0 7 .
Calculate the final value: 5 ⋅ 1 0 7 = 50 , 000 , 000 . The final answer is 50 , 000 , 000 .
Explanation
Understanding the Problem We are asked to find the value of the expression 0. 2 6 "."5 0 7 . To make it easier to work with, we can rewrite 0.2 as a fraction and 50 as a product of its prime factors.
Rewriting the Expression We can rewrite 0.2 as 5 1 and 50 as 5 ⋅ 10 . Now, we substitute these values into the expression: ( 5 1 ) 6 ⋅ ( 5 ⋅ 10 ) 7
Applying Exponent Rules Next, we apply the exponent rules to simplify the expression. Recall that ( ab ) n = a n b n and ( b a ) n = b n a n . Thus, we have 5 6 1 ⋅ 5 7 ⋅ 1 0 7 We can rewrite this as 5 7 − 6 ⋅ 1 0 7 = 5 1 ⋅ 1 0 7 = 5 ⋅ 1 0 7
Calculating the Final Value Now, we calculate 1 0 7 , which is 10 , 000 , 000 . Then, we multiply 5 by 10 , 000 , 000 to get the final answer: 5 ⋅ 10 , 000 , 000 = 50 , 000 , 000
Final Answer Therefore, the value of the expression 0. 2 6 ⋅ 5 0 7 is 50 , 000 , 000 .
Examples
This type of calculation is useful in various fields, such as finance and science, where exponential growth or decay is involved. For example, calculating the future value of an investment that grows at a certain percentage annually involves exponential calculations. Similarly, in physics, calculating the decay of radioactive materials involves exponential functions. Understanding how to manipulate and simplify exponential expressions is crucial for making accurate predictions and informed decisions in these areas.
The value of the expression 0. 2 6 ⋅ 5 0 7 simplifies to 50 , 000 , 000 by rewriting 0.2 and 50 in fraction and product form, respectively, and applying exponent rules. After simplification and calculation, we find the final result. Hence, the answer is 50 , 000 , 000 .
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