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Find the value of six cube root of 448 minus six cube root of 56 expressing your answer in the simplest form.
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So to help us solve this problem, we’ve got a rule that we can remember.
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That’s if we have the cube root of 𝑎𝑏, it’s equal to the cube root of 𝑎 multiplied by the cube root of 𝑏.
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This will also work if we have the cube root of 𝑎𝑏𝑐, this will be equal to the cube root of 𝑎 multiplied by the cube root of 𝑏 multiplied by the cube root of 𝑐.
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So what we need to do is actually divide up our numbers within the cube root into cube numbers and another number to see if it can help us to simplify.
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So we’re gonna start with 448, as this is in our first term.
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So how am I gonna simplify this?
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How can I break this down into cube numbers?
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Well, what I’m gonna do is actually use prime factor decomposition to break it down into its prime factors cause this will be able to help us.
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So if we’re gonna divide 448 into its prime factors, what we’re gonna do is divide it by a prime number.
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And the first prime number I’ve decided to divide it by is two cause it is nice and easy.
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So what we do is we divide 448 by two.
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So we get two and 224.
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And I’ve circled the two because that’s a prime number.
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And then, again, I’ve divided by two.
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So this time, 224 divided by two is 112 and by two again is 56.
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And now, I’ve actually decided to divide it by seven and that’s cause seven is a prime number.
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And actually, if we divide 56 by seven, we get seven and eight.
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And also, if we get two multiplied by two multiplied by two, so our first three prime factors, this will also give us eight.
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And eight is a cube number.
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So this can be really useful for the problem we’re solving cause we’re looking at cube root.
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So therefore, we can say that 448 is equal to eight multiplied by eight multiplied by seven.
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Okay, so now, let’s put this back into our original expression to see if it will help us.
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Well, now, if we rewrite six cube root of 448 minus six cube root of 56, we get six cube root of eight multiplied by eight multiplied by seven minus six cube root of eight multiplied by seven.
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Again, that’s because 56 is eight multiplied by seven.
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So therefore, using the rule that we started off with, we can say this is six cube root of eight multiplied by cube root of eight multiplied by cube root of seven minus six cube root of eight multiplied by cube root of seven.
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Well, we know that the cube root of eight is equal to two.
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And that’s because we said eight is a cube number because two multiplied by two multiplied by two is equal to eight.
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So therefore, we’ve got six multiplied by two multiplied by two multiplied by cube root of seven minus six multiplied by two multiplied by the cube root of seven.
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So we’re gonna get 24 cube root of seven and that’s cause six multiplied by two is 12 multiplied by another two is 24 minus 12 cube root of seven and that’s cause six multiplied by two is 12.
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So we’ve got 24 cube root of seven minus 12 cube root of seven, which will give us a final answer of 12 cube root of seven.
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So therefore, we can say that the value of six cube root of 448 minus six cube root of 56 expressed in the simplest form is 12 cube root seven.