Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Sure! Let's go through the steps to understand and solve the problem.
Given:
[tex]\[ x = 7 + 5\sqrt{2} \][/tex]
We need to determine [tex]\( \sqrt[3]{x} - 1 \)[/tex].
1. Calculate [tex]\( x \)[/tex]:
Since [tex]\( x = 7 + 5\sqrt{2} \)[/tex], we already have it in its defined form.
- [tex]\( 7 \)[/tex] is a rational number.
- [tex]\( 5\sqrt{2} \)[/tex] contains an irrational part because [tex]\( \sqrt{2} \)[/tex] is an irrational number.
So, [tex]\( x \)[/tex] is a combination of a rational and an irrational number, which means [tex]\( x \)[/tex] itself is an irrational number.
2. Find the cube root of [tex]\( x \)[/tex]:
To find [tex]\( \sqrt[3]{x} \)[/tex], we consider the cube root of the irrational number [tex]\( x \)[/tex].
3. Subtract 1 from the cube root of [tex]\( x \)[/tex]:
Let [tex]\( y = \sqrt[3]{x} - 1 \)[/tex]
Since [tex]\( x \)[/tex] is irrationational:
- The cube root [tex]\( \sqrt[3]{x} \)[/tex] is also an irrational number, as the cube root of an irrational number often remains irrational.
- Subtracting 1 from an irrational number still keeps it irrational.
Thus, [tex]\( y \)[/tex] is an irrational number, which means that [tex]\( \sqrt[3]{x} - 1 \)[/tex] is a surd.
Given Answer: After going through the provided steps, we find that the calculated numerical value for [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \)[/tex] is approximately [tex]\( 1.414213562373095 \)[/tex].
Since the result [tex]\( 1.414213562373095 \)[/tex] is the value for [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \)[/tex], let's analyze it in context of the options provided:
1. Whether [tex]\( 1.414213562373095 \approx \sqrt{3} \)[/tex]?
- No, because [tex]\( \sqrt{3} \approx 1.732 \)[/tex]
2. Whether [tex]\( 1.414213562373095 \approx \sqrt{5} \)[/tex]?
- No, because [tex]\( \sqrt{5} \approx 2.236 \)[/tex]
Finally, the result of [tex]\( \sqrt[3]{x} - 1 \)[/tex] was shown to be irrational, denoted as a surd here.
Conclusion: Neither [tex]\( \sqrt{3} \)[/tex] nor [tex]\( \sqrt{5} \)[/tex] matches the obtained result exactly.
Therefore, the correct conclusion is:
- [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \approx 1.414213562373095 \)[/tex]
- And neither [tex]\( \sqrt{3} \)[/tex] nor [tex]\( \sqrt{5} \)[/tex] are the exact matches. Both numbers mentioned are indeed surds.
Given:
[tex]\[ x = 7 + 5\sqrt{2} \][/tex]
We need to determine [tex]\( \sqrt[3]{x} - 1 \)[/tex].
1. Calculate [tex]\( x \)[/tex]:
Since [tex]\( x = 7 + 5\sqrt{2} \)[/tex], we already have it in its defined form.
- [tex]\( 7 \)[/tex] is a rational number.
- [tex]\( 5\sqrt{2} \)[/tex] contains an irrational part because [tex]\( \sqrt{2} \)[/tex] is an irrational number.
So, [tex]\( x \)[/tex] is a combination of a rational and an irrational number, which means [tex]\( x \)[/tex] itself is an irrational number.
2. Find the cube root of [tex]\( x \)[/tex]:
To find [tex]\( \sqrt[3]{x} \)[/tex], we consider the cube root of the irrational number [tex]\( x \)[/tex].
3. Subtract 1 from the cube root of [tex]\( x \)[/tex]:
Let [tex]\( y = \sqrt[3]{x} - 1 \)[/tex]
Since [tex]\( x \)[/tex] is irrationational:
- The cube root [tex]\( \sqrt[3]{x} \)[/tex] is also an irrational number, as the cube root of an irrational number often remains irrational.
- Subtracting 1 from an irrational number still keeps it irrational.
Thus, [tex]\( y \)[/tex] is an irrational number, which means that [tex]\( \sqrt[3]{x} - 1 \)[/tex] is a surd.
Given Answer: After going through the provided steps, we find that the calculated numerical value for [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \)[/tex] is approximately [tex]\( 1.414213562373095 \)[/tex].
Since the result [tex]\( 1.414213562373095 \)[/tex] is the value for [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \)[/tex], let's analyze it in context of the options provided:
1. Whether [tex]\( 1.414213562373095 \approx \sqrt{3} \)[/tex]?
- No, because [tex]\( \sqrt{3} \approx 1.732 \)[/tex]
2. Whether [tex]\( 1.414213562373095 \approx \sqrt{5} \)[/tex]?
- No, because [tex]\( \sqrt{5} \approx 2.236 \)[/tex]
Finally, the result of [tex]\( \sqrt[3]{x} - 1 \)[/tex] was shown to be irrational, denoted as a surd here.
Conclusion: Neither [tex]\( \sqrt{3} \)[/tex] nor [tex]\( \sqrt{5} \)[/tex] matches the obtained result exactly.
Therefore, the correct conclusion is:
- [tex]\( \sqrt[3]{7 + 5\sqrt{2}} - 1 \approx 1.414213562373095 \)[/tex]
- And neither [tex]\( \sqrt{3} \)[/tex] nor [tex]\( \sqrt{5} \)[/tex] are the exact matches. Both numbers mentioned are indeed surds.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
