Answer :
To determine if the product [tex]\(157 \times \sqrt{24}\)[/tex] is rational or irrational, let's look at the components involved.
1. Understanding the Expression:
- The expression given is [tex]\(157 \times \sqrt{24}\)[/tex].
2. Simplifying the Square Root:
- [tex]\(\sqrt{24}\)[/tex] can be broken down into its factors: [tex]\(24 = 4 \times 6\)[/tex].
- Therefore, [tex]\(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \times \sqrt{6}\)[/tex].
3. Calculating the Result:
- The value of [tex]\(\sqrt{24}\)[/tex] approximately equals 4.8989795.
4. Multiplying by 157:
- We calculate [tex]\(157 \times \sqrt{24}\)[/tex], which equals approximately 769.1397792.
5. Determining the Nature of the Result:
- A rational number can be expressed as a fraction of two integers. However, [tex]\(\sqrt{24}\)[/tex] is an irrational number because its decimal form is non-repeating and non-terminating.
- Multiplying the rational number 157 by the irrational number [tex]\(\sqrt{24}\)[/tex] results in an irrational number.
Thus, the product [tex]\(157 \times \sqrt{24}\)[/tex] is irrational.
1. Understanding the Expression:
- The expression given is [tex]\(157 \times \sqrt{24}\)[/tex].
2. Simplifying the Square Root:
- [tex]\(\sqrt{24}\)[/tex] can be broken down into its factors: [tex]\(24 = 4 \times 6\)[/tex].
- Therefore, [tex]\(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \times \sqrt{6}\)[/tex].
3. Calculating the Result:
- The value of [tex]\(\sqrt{24}\)[/tex] approximately equals 4.8989795.
4. Multiplying by 157:
- We calculate [tex]\(157 \times \sqrt{24}\)[/tex], which equals approximately 769.1397792.
5. Determining the Nature of the Result:
- A rational number can be expressed as a fraction of two integers. However, [tex]\(\sqrt{24}\)[/tex] is an irrational number because its decimal form is non-repeating and non-terminating.
- Multiplying the rational number 157 by the irrational number [tex]\(\sqrt{24}\)[/tex] results in an irrational number.
Thus, the product [tex]\(157 \times \sqrt{24}\)[/tex] is irrational.
