Let's solve the given expression, [tex]\( 4 \sqrt{2} + 5 \sqrt{4} \)[/tex], step by step to determine if the result is rational or irrational.
1. Evaluate the square roots:
- Calculate [tex]\( \sqrt{2} \)[/tex]. [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex]. Since [tex]\(\sqrt{2}\)[/tex] is not a perfect square, it is an irrational number.
- Calculate [tex]\( \sqrt{4} \)[/tex]. [tex]\(\sqrt{4} = 2\)[/tex]. Since [tex]\(\sqrt{4}\)[/tex] is a perfect square, it is a rational number.
2. Multiply by the coefficients:
- Multiply [tex]\( 4 \)[/tex] by [tex]\( \sqrt{2} \)[/tex].
[tex]\[
4 \sqrt{2} \approx 4 \times 1.4142135623730951 = 5.656854249492381
\][/tex]
- Multiply [tex]\( 5 \)[/tex] by [tex]\( \sqrt{4} \)[/tex].
[tex]\[
5 \sqrt{4} = 5 \times 2 = 10
\][/tex]
3. Sum the results:
- Add the two products together.
[tex]\[
4 \sqrt{2} + 5 \sqrt{4} \approx 5.656854249492381 + 10 = 15.65685424949238
\][/tex]
4. Determine the nature of the result:
- [tex]\( 4 \sqrt{2} \)[/tex] is an irrational number.
- [tex]\( 10 \)[/tex] is a rational number.
- Sum of an irrational number and a rational number is always an irrational number.
Therefore, the final answer [tex]\( 15.65685424949238 \)[/tex] is irrational because it includes an irrational component, specifically [tex]\( 4 \sqrt{2} \)[/tex].
