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.
