To solve the expression [tex]\(157 \times \sqrt{24}\)[/tex], follow these steps:
1. Identify the components:
- The number 157 is a constant.
- [tex]\(\sqrt{24}\)[/tex] represents the square root of 24.
2. Simplify the square root (if possible):
- The number 24 can be broken down into its prime factors: [tex]\(24 = 2^3 \times 3\)[/tex].
- The square root of 24, [tex]\(\sqrt{24}\)[/tex], can be simplified to [tex]\(\sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\)[/tex].
3. Combine the simplified terms:
- Substitute the simplified form back into the expression: [tex]\(157 \times \sqrt{24} = 157 \times 2\sqrt{6} = 314\sqrt{6}\)[/tex].
4. Calculate the numerical value:
- Using a calculator, determine the value of [tex]\(\sqrt{6}\)[/tex] which is approximately 2.44949.
- Multiply this approximate value by 314: [tex]\(314 \times 2.44949 \approx 769.1397792339178\)[/tex].
Thus, the value of [tex]\(157 \times \sqrt{24}\)[/tex] is approximately 769.1397792339178.
Now, let's classify the result as rational or irrational:
- The product of a rational number (157) and an irrational number ([tex]\(\sqrt{24}\)[/tex]) is always irrational.
Therefore, [tex]\(157 \times \sqrt{24}\)[/tex] is an irrational number. The approximate value is [tex]\(769.1397792339178\)[/tex].
