To solve the equation \( 2^{1/2} - 2^{1/4} d = \frac{7^{9/16}} \), let's first simplify the equation step by step.
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Rewrite the exponents:
- \( 2^{1/2} \) is the same as \( \sqrt{2} \).
- \( 2^{1/4} \) is the same as \( \sqrt[4]{2} \).
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Rearrange the equation: \[ 2^{1/2} - \frac{7^{9/16}}{2^{1/4}} = d \]
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Find \( 2^{1/2} \) and \( 2^{1/4} \):
- \( 2^{1/2} = \sqrt{2} \).
- \( 2^{1/4} = \sqrt[4]{2} \).
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Convert the equation: We can express everything using approximate numerical values or just keep them in terms of exponents.
Now, without full numerical simplifications, let’s isolate \( d \):
\[ d = 2^{1/2} - 2^{1/4} d \implies d + 2^{1/4} d = 2^{1/2} \] \[ d(1 + 2^{1/4}) = 2^{1/2} \] \[ d = \frac{2^{1/2}}{1 + 2^{1/4}} \]
Next, to find \( d \) explicitly using numerical values if needed, you can calculate \( 2^{1/2} \), \( 2^{1/4} \), and substitute to find \( d \).
Assuming you meant to find just \( d \), we can isolate it or evaluate \( d \) numerically based on real values:
- Approximate \( 2^{1/2} \approx 1.4142 \)
- Approximate \( 2^{1/4} \approx 1.1892 \)
From here, simply substitute those values to compute \( d \):
\[ d \approx \frac{1.4142}{1 + 1.1892} \approx \frac{1.4142}{2.1892}. \] Now calculating it gives:
\( d \approx 0.6455 \).
To summarize, the value of \( d \) will present as:
\[ d \approx 0.6455. \] If you meant something different with the notation of the equation, please clarify further for specific steps!