Question
Sbuve that \( (2 \sqrt{2})^{5}=32 \sqrt{2} \)
Ask by Newton Tyler. in Myanmar
Dec 28,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (2 \sqrt{2})^{5} = 128 \sqrt{2} \)
Solution
It appears there may be a miscalculation in the statement \( (2 \sqrt{2})^{5} = 32 \sqrt{2} \). Let's carefully evaluate \( (2 \sqrt{2})^{5} \) step by step to find the correct result.
### Step 1: Express in Exponential Form
First, express each component in its exponential form using base 2:
\[
2 = 2^1 \quad \text{and} \quad \sqrt{2} = 2^{1/2}
\]
Therefore,
\[
2 \sqrt{2} = 2^1 \cdot 2^{1/2} = 2^{1 + 1/2} = 2^{3/2}
\]
### Step 2: Raise to the Power of 5
Now, raise \( 2^{3/2} \) to the 5th power:
\[
(2^{3/2})^{5} = 2^{(3/2) \times 5} = 2^{15/2}
\]
### Step 3: Simplify the Exponent
Break down the exponent \( \frac{15}{2} \):
\[
2^{15/2} = 2^{7 + 1/2} = 2^7 \cdot 2^{1/2} = 128 \cdot \sqrt{2}
\]
### Conclusion
Putting it all together:
\[
(2 \sqrt{2})^{5} = 128 \sqrt{2}
\]
Therefore, the correct evaluation is:
\[
(2 \sqrt{2})^{5} = 128 \sqrt{2}
\]
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Simplify this solution Bonus Knowledge
To prove that \( (2 \sqrt{2})^{5} = 32 \sqrt{2} \), let's start by simplifying the left-hand side: \[ (2 \sqrt{2})^{5} = 2^{5} \cdot (\sqrt{2})^{5} \] Now calculate each part: \[ 2^{5} = 32 \] \[ (\sqrt{2})^{5} = (2^{1/2})^{5} = 2^{5/2} = 2^{2} \cdot 2^{1/2} = 4 \sqrt{2} \] Now, combine both parts: \[ (2 \sqrt{2})^{5} = 32 \cdot 4 \sqrt{2} = 128 \sqrt{2} \] Actually, I made an error above because \( 32 \cdot 4 \sqrt{2} \) is not equal to \( 32 \sqrt{2} \) but rather \( 128 \sqrt{2} \). Thus, the original statement \( (2 \sqrt{2})^{5} = 32 \sqrt{2} \) is not correct since \( (2 \sqrt{2})^{5} = 128 \sqrt{2} \). So the equality does not hold.
