Answer :
To find out by how much the sum of \(3 \frac{2}{3}\) and \(2^{1/5}\) is less than 7, let's break down the steps carefully:
1. Convert the mixed number \(3 \frac{2}{3}\) to a decimal:
The mixed number \(3 \frac{2}{3}\) consists of a whole number part (3) and a fraction part (\(\frac{2}{3}\)).
- The fraction \(\frac{2}{3}\) can be converted to a decimal by dividing the numerator by the denominator: \(\frac{2}{3} \approx 0.6666666666666666\).
- Therefore, \(3 \frac{2}{3}\) as a decimal is \(3 + 0.6666666666666666 = 3.6666666666666665\).
2. Calculate \(2^{1/5}\):
The fifth root of 2, \(2^{1/5}\), can be approximated as:
[tex]\[ 2^{1/5} \approx 1.148698354997035 \][/tex]
3. Add \(3 \frac{2}{3}\) and \(2^{1/5}\):
Now, sum the two values:
[tex]\[ 3.6666666666666665 + 1.148698354997035 = 4.815365021663702 \][/tex]
4. Determine how much less this sum is than 7:
To find out how much less the sum is than 7, subtract the sum from 7:
[tex]\[ 7 - 4.815365021663702 = 2.184634978336298 \][/tex]
Therefore, the sum of [tex]\(3 \frac{2}{3}\)[/tex] and [tex]\(2^{1/5}\)[/tex] is approximately [tex]\(4.815365021663702\)[/tex], and it is approximately [tex]\(2.184634978336298\)[/tex] less than 7.
1. Convert the mixed number \(3 \frac{2}{3}\) to a decimal:
The mixed number \(3 \frac{2}{3}\) consists of a whole number part (3) and a fraction part (\(\frac{2}{3}\)).
- The fraction \(\frac{2}{3}\) can be converted to a decimal by dividing the numerator by the denominator: \(\frac{2}{3} \approx 0.6666666666666666\).
- Therefore, \(3 \frac{2}{3}\) as a decimal is \(3 + 0.6666666666666666 = 3.6666666666666665\).
2. Calculate \(2^{1/5}\):
The fifth root of 2, \(2^{1/5}\), can be approximated as:
[tex]\[ 2^{1/5} \approx 1.148698354997035 \][/tex]
3. Add \(3 \frac{2}{3}\) and \(2^{1/5}\):
Now, sum the two values:
[tex]\[ 3.6666666666666665 + 1.148698354997035 = 4.815365021663702 \][/tex]
4. Determine how much less this sum is than 7:
To find out how much less the sum is than 7, subtract the sum from 7:
[tex]\[ 7 - 4.815365021663702 = 2.184634978336298 \][/tex]
Therefore, the sum of [tex]\(3 \frac{2}{3}\)[/tex] and [tex]\(2^{1/5}\)[/tex] is approximately [tex]\(4.815365021663702\)[/tex], and it is approximately [tex]\(2.184634978336298\)[/tex] less than 7.
