Rewrite the expression using positive exponents only. (Simplify your answer completely.)
(x^3-y^3)(x^-3+y^-3)
To rewrite the expression using positive exponents only, we can simplify the individual terms within each set of parentheses and then multiply them together.
Let's simplify the first set of parentheses:
x^3 - y^3
Now, let's simplify the second set of parentheses:
x^-3 + y^-3
When we multiply these two simplified expressions together, we get:
(x^3 - y^3)(x^-3 + y^-3) = x^3 * x^-3 + x^3 * y^-3 - y^3 * x^-3 - y^3 * y^-3
Using the properties of exponents, we know that x^a * x^b = x^(a+b) and that a^(-n) = 1/a^n. Applying these properties, we can simplify the expression further:
= x^(3+(-3)) + x^3 * y^-3 - y^3 * x^-3 - y^(3+(-3))
= x^0 + x^3 * y^-3 - y^3 * x^-3 - y^0
= 1 + (x^3 / y^3) - (y^3 / x^3) - 1
= 1 + (x^3 / y^3) - (y^3 / x^3)
Thus, the expression (x^3 - y^3)(x^-3 + y^-3) simplified using positive exponents only is 1 + (x^3 / y^3) - (y^3 / x^3).
To rewrite the expression using positive exponents only, we can use the rule of negative exponents. The negative exponent rule states that any term with a negative exponent can be moved to the other side of the fraction and the exponent becomes positive.
Using this rule, let's rewrite the expression:
(x^3 - y^3)(x^-3 + y^-3)
Moved negative exponents to the other side:
(x^3 - y^3) / (x^3 + y^3)
Therefore, the simplified expression with positive exponents only is:
(x^3 - y^3) / (x^3 + y^3)
(x^3-y^3)(x^-3+y^-3)
= x^0 - x^-3y^3 + x^3y^-3 - y^0
= y^3/x^3 + x^3/y^3