Use the properties of exponents and radicals to generate an equivalent expression for (256√10)^5 / 512^1/3

Question

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Answer 1

To simplify the expression \((256\sqrt{10})^5 / 512^{1/3}\), we can use the properties of exponents and radicals.

Let's break it down step by step.

1. Simplify \((256\sqrt{10})^5\):

   • First, note that \(256 = 2^8\) and \(\sqrt{10} = 10^{1/2}\).

   • Therefore, we can rewrite \(256\sqrt{10}\) as: \[ 256\sqrt{10} = 2^8 \cdot 10^{1/2} \]

   • Now raise this to the 5th power: \[ (256\sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = 2^{8 \cdot 5} \cdot 10^{(1/2) \cdot 5} = 2^{40} \cdot 10^{5/2} \]

2. Simplify \(512^{1/3}\):

   • Next, recognize that \(512 = 2^9\).
   • Thus, we can write: \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8 \]

3. Combine the expression: Now we need to combine both parts: \[ \frac{(256\sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{2^3} = 2^{40-3} \cdot 10^{5/2} = 2^{37} \cdot 10^{5/2} \]

4. Final expression: Therefore, the equivalent expression for \(\frac{(256\sqrt{10})^5}{512^{1/3}}\) is: \[ 2^{37} \cdot 10^{5/2} \]

You might also present \(10^{5/2}\) in another form. For instance: \[ 10^{5/2} = \sqrt{10^5} = \sqrt{100000} = 100\sqrt{10} \] Thus alternatively, you could write the final expression as:

\[ 2^{37} \cdot 100\sqrt{10} \]

Both forms \(2^{37} \cdot 10^{5/2}\) and \(2^{37} \cdot 100\sqrt{10}\) are equivalent ways to express the original expression.

Answer 2

the correct answer is 2 why?

Answer 3

Let's go through the steps again and ensure we simplify the expression correctly:

We need to simplify \((256\sqrt{10})^5 / 512^{1/3}\).

1. Simplify \( (256\sqrt{10})^5 \):

   • First, we can express 256 as \( 2^8 \) and \(\sqrt{10}\) as \( 10^{1/2} \): \[ 256 = 2^8 \quad \text{and} \quad \sqrt{10} = 10^{1/2} \]
   • Therefore: \[ 256\sqrt{10} = 2^8 \cdot 10^{1/2} \]
   • Now, raising this to the 5th power: \[ (256\sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = 2^{8 \cdot 5} \cdot 10^{(1/2) \cdot 5} = 2^{40} \cdot 10^{5/2} \]

2. Simplify \( 512^{1/3} \):

   • We can express 512 as \( 2^9 \): \[ 512 = 2^9 \]
   • Therefore: \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8 \]

3. Combine the expression: Now we have: \[ \frac{(256\sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{2^3} \] This simplifies to: \[ 2^{40 - 3} \cdot 10^{5/2} = 2^{37} \cdot 10^{5/2} \]

4. Further Simplification:

   • We can express \( 10^{5/2} \) as \( 10^{2.5} = 10^2 \cdot 10^{0.5} = 100\sqrt{10} \). Thus, we can rewrite: \[ 2^{37} \cdot 10^{5/2} = 2^{37} \cdot 100\sqrt{10} \]

5. Equivalence to 2: If the final output should be 2, there might have been a flaw in interpretation. Let's check the numerical dimensionality:

   Since \( 10^{5/2} \), which represents a decimal value scaling based on base 10, results in significant shifts upward. Let's compute directly with defined values:

   1. Calculate \( 256\sqrt{10} \) then raise to 5 \[ \sqrt{10} \approx 3.16 \quad \therefore \quad 256\sqrt{10} \approx 256 \times 3.16 \approx 809.6 \] Raised to power 5 makes significant scales but then divided by 8 orders downwards.

   2. Since your goal is a simple integration to express formations—if we take flooring or rounding observed computations, they can approximate back toward small integers.

It would help if you clarify what context or specific conditions lead you towards seeing a derived equivalence of 2. Nowhere does our calculation collapse directly to 2 within the base evaluations I traversed without constraints. Double-check how results converge reflecting alternative methods or approximations could yield quicker results!