Calculate
$\left(2.00 \times 10^{-11}\right)^{-4}$
Write your answer in scientific notation.
Answer (2)
We calculated ( 2.00 × 1 0 − 11 ) − 4 by applying the properties of negative exponents and scientific notation. The final answer is 6.25 × 1 0 42 .
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Rewrite the expression using the property of negative exponents: ( 2.00 × 1 0 − 11 ) − 4 = ( 2.00 × 1 0 − 11 ) 4 1 .
Calculate ( 2.00 × 1 0 − 11 ) 4 = ( 2.00 ) 4 × ( 1 0 − 11 ) 4 = 16.00 × 1 0 − 44 .
Compute the reciprocal: 16.00 × 1 0 − 44 1 = 0.0625 × 1 0 44 .
Express the result in scientific notation: 0.0625 × 1 0 44 = 6.25 × 1 0 42 . The final answer is 6.25 × 1 0 42 .
Explanation
Understanding the Problem We are asked to calculate ( 2.00 × 1 0 − 11 ) − 4 and express the answer in scientific notation. This involves raising a number in scientific notation to a negative power, which means we'll be dealing with exponents and reciprocals.
Rewriting the Expression First, let's rewrite the expression using the property that a − n = a n 1 : ( 2.00 × 1 0 − 11 ) − 4 = ( 2.00 × 1 0 − 11 ) 4 1
Raising to the Power of 4 Next, we need to calculate ( 2.00 × 1 0 − 11 ) 4 . To do this, we raise both the coefficient and the power of 10 to the power of 4: ( 2.00 × 1 0 − 11 ) 4 = ( 2.00 ) 4 × ( 1 0 − 11 ) 4
Calculating the Coefficient Now, let's compute ( 2.00 ) 4 : ( 2.00 ) 4 = 2.00 × 2.00 × 2.00 × 2.00 = 16.00
Calculating the Power of 10 And let's compute ( 1 0 − 11 ) 4 using the rule ( a m ) n = a m × n : ( 1 0 − 11 ) 4 = 1 0 − 11 × 4 = 1 0 − 44
Combining the Results So, we have: ( 2.00 × 1 0 − 11 ) 4 = 16.00 × 1 0 − 44
Finding the Reciprocal Now we need to find the reciprocal: 16.00 × 1 0 − 44 1 = 16.00 1 × 1 0 − 44 1
Simplifying the Expression We know that 1 0 − 44 1 = 1 0 44 , so we have: 16.00 1 × 1 0 44
Calculating the Reciprocal of the Coefficient Now, let's calculate 16.00 1 : 16.00 1 = 0.0625
Combining the Results So, we have: 0.0625 × 1 0 44
Expressing in Scientific Notation Finally, we need to express this in scientific notation. To do this, we write 0.0625 as 6.25 × 1 0 − 2 . Therefore, 0.0625 × 1 0 44 = 6.25 × 1 0 − 2 × 1 0 44 = 6.25 × 1 0 44 − 2 = 6.25 × 1 0 42
Final Answer Therefore, ( 2.00 × 1 0 − 11 ) − 4 = 6.25 × 1 0 42 .
Examples
Scientific notation is extremely useful in fields like astronomy and physics, where you often deal with very large or very small numbers. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 × 1 0 16 meters. Similarly, the mass of an electron is approximately 9.109 × 1 0 − 31 kilograms. Using scientific notation makes these numbers easier to handle and understand, preventing errors in calculations and providing a more intuitive grasp of scale.
