SImplify:
(x - 9/x) / (1 + x/3)
and (m^-1 + n^-1) / (m^-3 + n^-3)
I have no idea what to do
Answer to #2:
m^4n^2.
I subtracted the m's and subtracted the m's. It's one of the exponent rules.
I have NO idea how to do the first question.
you cant do that, Miley, the m's and n's are separated by an addition sign, so you cant use the quotient of powers rule
oh okay sry, at least i tried
for the first one, just multiply them using FOIL
to get
x + (x^2)/3 - 9/x - 3
for the second, use the rule for negative exponents to write
(1/m + 1/n)/(1/m^3 + 1/n^3)
finding common denominators and simplifying both top and bottom I got
(m+n)/mn / (m^3 + n^3)/m^3n^3
= (m+n)/mn * (m^3n^3)/(m^3 + n^3)
= m^2n^2(m+n) / (m^3 + n^3)
the bottom factors to (m+n)(m^2 - mn + n^2)
so we have
m^2n^2(m+n) / ((m+n)(m^2 - mn + n^2))
= m^2n^2 / (m^2 - mn + n^2)
To simplify the expressions, we can start by simplifying each term separately and then combining like terms.
Expression 1: (x - 9/x) / (1 + x/3)
Step 1: Simplify the numerator term.
To simplify the term x - 9/x, we need to find a common denominator. The common denominator for the terms x and 9/x is x. So, we multiply the first term (x) by x/x to get (x^2), and the second term (9/x) by 9/9 to get (9/x^2). Now our numerator becomes (x^2 - 9/x^2).
Step 2: Simplify the denominator term.
The denominator term is 1 + x/3.
Step 3: Simplify the expression.
Divide the numerator and denominator by the greatest common factor (GCF) to further simplify the expression, if possible. In this case, there is no common factor between the numerator and denominator:
(x^2 - 9/x^2) / (1 + x/3)
Expression 2: (m^-1 + n^-1) / (m^-3 + n^-3)
Step 1: Simplify the numerator term.
The numerator term is (m^-1 + n^-1).
Step 2: Simplify the denominator term.
The denominator term is (m^-3 + n^-3).
Step 3: Simplify the expression.
We can use the rules of exponents to simplify the expression further. The rule for adding exponents is:
a^m + a^n = a^(m+n).
(m^-1 + n^-1) / (m^-3 + n^-3)
= (1/m + 1/n) / (1/m^3 + 1/n^3)
= (mn / (mn)) * (1/m + 1/n) / (1/m^3 + 1/n^3)
= (mn / (mn)) * (n + m) / (n^3 + m^3)
= (n + m) / (n^3 + m^3)
Therefore, the simplified expression for (m^-1 + n^-1) / (m^-3 + n^-3) is (n + m) / (n^3 + m^3).