Show that a^12 can be written as a perfect square and a perfect cube

Mathematics

1 answer:

Elden [556K]3 years ago

3 0

Answer:

Step-by-step explanation:

Doing this the long way, write a^12 as a product of "a"s

a^12 = a * a * a * a * a * a * a * a * a * a * a * a

Now group them into groups of 4

a^12 = a * a * a * a * a * a * a * a * a * a * a * a

There are 3 groups of 4 so you can write it as [(a^4)]^3 which is a cube

Now do the same thing with groups of 6s

a^12 = a * a * a * a * a * a * a * a * a * a * a * a

a^12 = (a^6)^2

So the two answers are

Perfect cube: [(a^4)]^3

Perfect square: (a^6)^2

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What value of cc will complete the square below and make the expression a perfect square trinomial?

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Question: What value of c will complete the square below ( $x^{2} +30x+c$ ) and make the expression a perfect square trinomial?

Answer: c = 225

Step-by-step explanation:

Perfect square trinomials come in the form a² + 2ab + b², which is equal to (a + b)². In the presented trinomial, we can immediately identify that a = x, and b² = c, but we need to find the numerical value of $c$ .

To do this, note that the middle term, or 2ab, corresponds with (is equal to) 30x. We know that a = x, and thus, 2ab = 2bx. Now, 2bx and 30x are corresponding terms; thus, 2bx = 30x.

Dividing by $2x$ on both sides gives us b = 15. Therefore, c = b² = 15² = 225. (As a squared binomial, this would be (x + 15)² as a = x and b = 15.)