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Find the sum.

28 cube root of 5 end root minus 5 cube root of negative 320 end root

5 answers

First, we will simplify each term in the expression \( 28 \sqrt[3]{5} - 5 \sqrt[3]{-320} \).

  1. First term: The first term is \( 28 \sqrt[3]{5} \).

  2. Second term: For the second term \( -5 \sqrt[3]{-320} \), we can simplify \( \sqrt[3]{-320} \). \[ -320 = -1 \times 320 \] We can take the cube root: \[ \sqrt[3]{-320} = \sqrt[3]{-1} \times \sqrt[3]{320} = -\sqrt[3]{320} \] Therefore, \[ -5 \sqrt[3]{-320} = -5 \times (-\sqrt[3]{320}) = 5 \sqrt[3]{320} \]

Next, we simplify \( 320 \): \[ 320 = 32 \times 10 = 2^5 \times 10 = 2^5 \times 2 \times 5 = 2^6 \times 5 \] Thus, \[ \sqrt[3]{320} = \sqrt[3]{2^6 \times 5} = \sqrt[3]{(2^2)^3 \times 2 \times 5} = 2^2 \times \sqrt[3]{10} = 4 \sqrt[3]{10} \] So \( 5 \sqrt[3]{320} \) becomes: \[ 5 \sqrt[3]{320} = 5 \times 4 \sqrt[3]{10} = 20 \sqrt[3]{10} \]

Now we substitute back: \[ 28 \sqrt[3]{5} + 20 \sqrt[3]{10} \]

At this point, we analyze whether further simplification is possible. The terms \( \sqrt[3]{5} \) and \( \sqrt[3]{10} = \sqrt[3]{2 \times 5} = \sqrt[3]{2} \cdot \sqrt[3]{5} \) hint that we can express \( \sqrt[3]{10} \) in terms of \( \sqrt[3]{5} \): \[ 20 \sqrt[3]{10} = 20 \cdot \sqrt[3]{2} \cdot \sqrt[3]{5} \]

So, adding the two terms together, we have: \[ 28 \sqrt[3]{5} + 20 \sqrt[3]{10} = 28 \sqrt[3]{5} + 20 \cdot \sqrt[3]{2} \cdot \sqrt[3]{5} = (28 + 20 \sqrt[3]{2}) \sqrt[3]{5} \]

The final expression becomes: \[ \boxed{(28 + 20 \sqrt[3]{2}) \sqrt[3]{5}} \]

no slashes or special characters please

Certainly! Let's go through the problem without using any special characters or slashes.

We need to simplify and find the sum of:

28 times the cube root of 5 minus 5 times the cube root of negative 320.

First, simplify each part.

  1. The first term is simply: 28 times the cube root of 5.

  2. For the second term, we need to simplify: negative 5 times the cube root of negative 320.

The cube root of negative 320 can be simplified as: The cube root of negative 320 is equal to the cube root of negative one multiplied by the cube root of 320. This gives us negative cube root of 320.

Now, we rewrite the second term as: negative 5 times negative cube root of 320, which equals 5 times cube root of 320.

Next, we will simplify cube root of 320. 320 can be factored as 32 times 10, and since 32 is equal to 2 to the power of 5, we get: 320 equals 2 to the power of 5 times 10, or 2 to the power of 5 times 2 times 5, which is 2 to the power of 6 times 5.

Thus, we can find the cube root of 320 as: cube root of 320 equals cube root of 2 to the power of 6 times 5. This can be simplified to: the cube root of (2 squared cubed times 2 times 5), which gives us: 2 squared times cube root of 10, or 4 times cube root of 10.

Now we can substitute back: The second term, which is 5 times cube root of 320, becomes: 5 times 4 times cube root of 10, which equals 20 times cube root of 10.

Now combining the terms: The expression becomes: 28 times cube root of 5 plus 20 times cube root of 10.

To summarize: The final result is: 28 times cube root of 5 plus 20 times cube root of 10.

This is the simplified form of the original expression.

If you need the final answer expressed as one term, it could be written as: (28 plus 20 times cube root of 2) times cube root of 5.

So the result is: (28 plus 20 times cube root of 2) times cube root of 5.

So, how do you write the answer in a number?! What's the answer! Don't use words, use numbers for the answer!!

The answer is:

28 * (cube root of 5) + 20 * (cube root of 10)

Which is approximately:

69.499