List prime numbers between 100 and 120 and find the product of the largest and smallest. Classify 1, 2, and…

Question

List prime numbers between 100 and 120 and find the product of the largest and smallest. Classify 1, 2, and sqrt(5) as rational or irrational. Calculate the fraction of journey covered.

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6. Prime numbers and their product

List prime numbers between 100 and 120.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The prime numbers between 100 and 120 are:
$101, 103, 107, 109, 113$
Identify the largest and smallest prime numbers.
Smallest prime number = 101
Largest prime number = 113
Calculate the product of the largest and smallest prime numbers.
Product = $101 \times 113$
$101 \times 113 = 11413$
The product of the largest and smallest prime numbers is $\boxed{11413}$ .

7. Classification of numbers
Define rational and irrational numbers.
A rational number can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$ .
An irrational number cannot be expressed as a simple fraction; its decimal representation is non-terminating and non-repeating.
Classify the given numbers.
- 1: This can be written as $\frac{1}{1}$ . Therefore, 1 is a rational number.
- 2: This can be written as $\frac{2}{1}$ . Therefore, 2 is a rational number.
- $\sqrt{5}$ : The square root of 5 is approximately 2.2360679..., which is a non-terminating and non-repeating decimal. Therefore, $\sqrt{5}$ is an irrational number.
8. Fraction of journey covered
Determine the total time for the journey.
Total time = 5 hours.
Determine the fraction of the journey covered in 1 hour.
Fraction covered in 1 hour = $\frac{1 \text{ hour}}{5 \text{ hours}} = \frac{1}{5}$ .
Use reciprocal tables to find the value of $\frac{1}{5}$ .
From reciprocal tables, the reciprocal of 5 is 0.2.
$\frac{1}{5} = 0.2$
The fraction of the journey covered in 1 hour is $\boxed{0.2}$ .

9. Evaluation using mathematical tables
Evaluate $\frac{1}{0.981}$ using reciprocal tables.
From reciprocal tables, $\frac{1}{0.981} \approx 1.01936799...$
Rounding to 5 decimal places for intermediate calculation: 1.01937
Evaluate $\frac{1}{9.81}$ using reciprocal tables.
From reciprocal tables, $\frac{1}{9.81} \approx 0.101936799...$
Rounding to 5 decimal places for intermediate calculation: 0.10194
Add the results.
$1.01937 + 0.10194 = 1.12131$
The value is approximately $\boxed{1.1213}$ .

10. Laws of indices
Rewrite the expression using fractional exponents.
$\sqrt[3]{\frac{256}{81}} = \left(\frac{256}{81}\right)^{\frac{1}{3}}$
Express the numbers as powers of their prime factors.
$256 = 2^8$
$81 = 3^4$
Substitute these into the expression.
$\left(\frac{2^8}{3^4}\right)^{\frac{1}{3}} = \frac{(2^8)^{\frac{1}{3}}}{(3^4)^{\frac{1}{3}}} = \frac{2^{\frac{8}{3}}}{3^{\frac{4}{3}}}$
Simplify the exponents.
$2^{\frac{8}{3}} = 2^{2 + \frac{2}{3}} = 2^2 \cdot 2^{\frac{2}{3}} = 4 \cdot \sqrt[3]{2^2} = 4 \sqrt[3]{4}$ $3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3^1 \cdot 3^{\frac{1}{3}} = 3 \sqrt[3]{3}$
Combine the terms.
$\frac{4 \sqrt[3]{4}}{3 \sqrt[3]{3}}$
To rationalize the denominator, multiply the numerator and denominator by $\sqrt[3]{3^2} = \sqrt[3]{9}$ .
$\frac{4 \sqrt[3]{4}}{3 \sqrt[3]{3}} \times \frac{\sqrt[3]{9}}{\sqrt[3]{9}} = \frac{4 \sqrt[3]{4 \times 9}}{3 \sqrt[3]{3 \times 9}} = \frac{4 \sqrt[3]{36}}{3 \sqrt[3]{27}} = \frac{4 \sqrt[3]{36}}{3 \times 3} = \frac{4 \sqrt[3]{36}}{9}$
The simplified expression is $\boxed{\frac{4 \sqrt[3]{36}}{9}}$ .

11. Evaluation using logarithm tables
Let $x = \frac{0.724 \times 20.37}{608}$ .
Take the logarithm base 10 of both sides.
$\log_{10} x = \log_{10} \left(\frac{0.724 \times 20.37}{608}\right)$
Apply logarithm rules.
$\log_{10} x = \log_{10} (0.724) + \log_{10} (20.37) - \log_{10} (608)$
Find the logarithms of each number using tables.
- $\log_{10} (0.724)$ : Characteristic is $\bar{1}$ . Mantissa of 724 is 0.8597. So, $\log_{10} (0.724) = \bar{1}.8597$ .
- $\log_{10} (20.37)$ : Characteristic is 1. Mantissa of 2037 is 0.3090. So, $\log_{10} (20.37) = 1.3090$ .
- $\log_{10} (608)$ : Characteristic is 2. Mantissa of 608 is 0.7839. So, $\log_{10} (608) = 2.7839$ .
Perform the addition and subtraction.
$\log_{10} x = \bar{1}.8597 + 1.3090 - 2.7839$ $\log_{10} x = ( -1 + 0.8597 ) + ( 1 + 0.3090 ) - ( 2 + 0.7839 )$ $\log_{10} x = -1 + 0.8597 + 1 + 0.3090 - 2 - 0.7839$ $\log_{10} x = (-1 + 1 - 2) + (0.8597 + 0.3090 - 0.7839)$ $\log_{10} x = -2 + (1.1687 - 0.7839)$ $\log_{10} x = -2 + 0.3848$ $\log_{10} x = \bar{2}.3848$
Find the antilogarithm of $\bar{2}.3848$ .
The mantissa is $0.3848

List prime numbers between 100 and 120 and find the product of the largest and smallest. Classify 1, 2, and sqrt(5) as rational or irrational. Calculate the fraction of journey covered.

Step-by-step solution

List prime numbers between 100 and 120.

Identify the largest and smallest prime numbers.

Calculate the product of the largest and smallest prime numbers.

Define rational and irrational numbers.

Classify the given numbers.

Determine the total time for the journey.

Determine the fraction of the journey covered in 1 hour.

Use reciprocal tables to find the value of $\frac{1}{5}$ .

Evaluate $\frac{1}{0.981}$ using reciprocal tables.

Evaluate $\frac{1}{9.81}$ using reciprocal tables.

Add the results.

Rewrite the expression using fractional exponents.

Express the numbers as powers of their prime factors.

Substitute these into the expression.

Simplify the exponents.

Combine the terms.

Take the logarithm base 10 of both sides.

Apply logarithm rules.

Find the logarithms of each number using tables.

Perform the addition and subtraction.

Find the antilogarithm of $\bar{2}.3848$ .

Still stuck on this one?

More Mathematics Questions

List prime numbers between 100 and 120 and find the product of the largest and smallest. Classify 1, 2, and sqrt(5) as rational or irrational. Calculate the fraction of journey covered.

Step-by-step solution

List prime numbers between 100 and 120.

Identify the largest and smallest prime numbers.

Calculate the product of the largest and smallest prime numbers.

Define rational and irrational numbers.

Classify the given numbers.

Determine the total time for the journey.

Determine the fraction of the journey covered in 1 hour.

Use reciprocal tables to find the value of 15\frac{1}{5}51​.

Evaluate 10.981\frac{1}{0.981}0.9811​ using reciprocal tables.

Evaluate 19.81\frac{1}{9.81}9.811​ using reciprocal tables.

Add the results.

Rewrite the expression using fractional exponents.

Express the numbers as powers of their prime factors.

Substitute these into the expression.

Simplify the exponents.

Combine the terms.

Take the logarithm base 10 of both sides.

Apply logarithm rules.

Find the logarithms of each number using tables.

Perform the addition and subtraction.

Find the antilogarithm of 2ˉ.3848\bar{2}.38482ˉ.3848.

Still stuck on this one?

More Mathematics Questions

Use reciprocal tables to find the value of $\frac{1}{5}$ .

Evaluate $\frac{1}{0.981}$ using reciprocal tables.

Evaluate $\frac{1}{9.81}$ using reciprocal tables.

Find the antilogarithm of $\bar{2}.3848$ .