Expand and combine like terms.
$(9 c^2+c^6)(9 c^2-c^6)=$
$\square$
Answer (1)
The problem requires expanding and simplifying the expression ( 9 c 2 + c 6 ) ( 9 c 2 − c 6 ) .
Recognize the expression as a difference of squares: ( a + b ) ( a − b ) = a 2 − b 2 .
Substitute a = 9 c 2 and b = c 6 into the formula.
Simplify each term: ( 9 c 2 ) 2 = 81 c 4 and ( c 6 ) 2 = c 12 .
Combine the simplified terms: 81 c 4 − c 12 .
Explanation
Recognizing the Pattern We are given the expression ( 9 c 2 + c 6 ) ( 9 c 2 − c 6 ) and asked to expand and simplify it. This looks like a difference of squares, which is a pattern that can help us simplify the expression.
Applying Difference of Squares The expression is in the form of ( a + b ) ( a − b ) where a = 9 c 2 and b = c 6 . We can use the difference of squares formula to expand the expression. The difference of squares formula is ( a + b ) ( a − b ) = a 2 − b 2 .
Substitution Now, let's substitute a = 9 c 2 and b = c 6 into the formula: ( 9 c 2 + c 6 ) ( 9 c 2 − c 6 ) = ( 9 c 2 ) 2 − ( c 6 ) 2
Simplifying the First Term Next, we simplify each term. First, we simplify ( 9 c 2 ) 2 . Remember that ( x y ) n = x n y n , so we have: ( 9 c 2 ) 2 = 9 2 ( c 2 ) 2 = 81 c 2 × 2 = 81 c 4
Simplifying the Second Term Now, we simplify ( c 6 ) 2 . Recall the power of a power rule: ( x m ) n = x m × n . Therefore: ( c 6 ) 2 = c 6 × 2 = c 12
Combining Terms Finally, we combine the simplified terms to get the final expression: 81 c 4 − c 12
Examples
Understanding how to expand and simplify expressions like this is useful in many areas of math, such as when you're solving equations or working with polynomials. For example, if you were designing a rectangular garden where the length is ( 9 c 2 + c 6 ) meters and the width is ( 9 c 2 − c 6 ) meters, the area of the garden would be ( 9 c 2 + c 6 ) ( 9 c 2 − c 6 ) square meters. Simplifying this expression to 81 c 4 − c 12 would give you a more concise way to calculate the garden's area for different values of c .
