Exercise 8 Page 60

Look closely at the expression 256n^4-c^4. It can be expressed as the difference of two perfect squares.

256n^4-c^4

Write as a power

16^2(n^2)^2-(c^2)^2

a^m b^m = (a b)^m

(16n^2)^2-(c^2)^2

Recall the formula to factor a difference of squares. a^2- b^2 ⇔ ( a+ b)( a- b) We can apply this formula to our expression. ( 16n^2)^2-( c^2)^2 ⇕ ( 16n^2+ c^2)( 16n^2- c^2) Now, let's take a look at the expression 16n^2-c^2. Once again, we can express it as a difference of two perfect squares. Let's do it!

16n^2-c^2

WritePow

Write as a power

4^2n^2-c^2

ProdPowII

a^m b^m = (a b)^m

(4n)^2-c^2

Now, we can apply the above formula to factor our expression completely. (16n^2+c^2)(( 4n)^2- c^2) ⇕ (16n^2+c^2)( 4n+ c)( 4n- c)

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We can apply the Distributive Property and compare the result with the given expression.

(16n^2 + c^2) (4n + c)(4n - c)

Distr

Distribute (4n+c)

(16n^2 + c^2) (4n(4n + c) - c(4n+c))

▼

Distribute 4n & - c

Distr

Distribute 4n

(16n^2 + c^2) (16n^2 + 4cn - c(4n+c))

Distr

Distribute - c

(16n^2 + c^2) (16n^2 + 4cn - 4cn - c^2)

SubTerm

Subtract term

(16n^2 + c^2) (16n^2 - c^2)

Distr

Distribute (16n^2 + c^2)

16n^2 (16n^2 + c^2 ) - c^2 (16n^2 + c^2)

▼

Distribute 16n^2 & - c^2

Distr

Distribute 16n^2

256n^4 + 16c^2n^2 - c^2 (16n^2 + c^2)

Distr

Distribute - c^2

256n^4 + 16c^2n^2 - 16c^2n^2 - c^4

SubTerm

Subtract term

256n^4 - c^4

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

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