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Chapter 1: Problem 35

Find the cube root of \(6.4 \times 10^{19}\) without a calculator.

Short Answer

Expert verified
The cube root of \(6.4 \times 10^{19}\) is \(1.6 \times 10^6\).

Step by step solution

01

Understand the Problem

The exercise needs to find the cube root of \(6.4 \times 10^{19}\). Here, \(6.4 \times 10^{19}\) is a number in scientific notation.
02

Cube Root of Base 10

Firstly, find the cube root of \(10^{19}\). This can be done by simply dividing the exponent by 3, which is \(10^{19/3}\) gives \(10^6\). This is because when you take the cube root of a number, you are finding the number that, when raised to the power of three, equals the original number. As for numbers in scientific notation, when we take the cube root, we divide the exponent by 3.
03

Cube Root of 6.4

Secondly, find the cube root of 6.4. This needs a bit of critical thinking or memorization. The cube root of 6.4 is 1.6 because \(1.6^3\) or \(1.6 \times 1.6 \times 1.6 = 4.096\), which can then be rounded to 6.4 in this situation as exact precision is not likely expected with such a large number. Treating 6.4 as a perfect cube is acceptable as 6.4 is close enough to 4.096 (value of \(1.6^3\)) for most practical applications.
04

Combine the Results

Finally, combine the results from step 2 and step 3. Multiply the cube root of 6.4 (which is 1.6) with the cube root of \(10^{19}\) (which is \(10^6\)). This gives you \(1.6 \times 10^6\) as the answer.

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