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Calculate the LCM of 2 and 6, and simplify the following fractions: 8/5, n/(n-2) (with n<2), and (n+2)/(n2-1) (with n<1).

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2 parts
a

6\text{6}

b

85\frac{8}{5}

nn−2\frac{n}{n-2}

n+2(n−1)(n+1)\frac{n+2}{(n-1)(n+1)}

Step-by-step solution

7 steps

Part a· 2 steps

احسب (Calculate)

The question asks to calculate for the numbers 2 and 6. Assuming it asks for the Least Common Multiple (LCM) of 2 and 6.

  1. Find the prime factorization of each number.

    2=212 = 2^1 6=21×316 = 2^1 \times 3^1
  2. Determine the LCM by taking the highest power of all prime factors present.

    LCM(2,6)=21×31=6\text{LCM}(2, 6) = 2^1 \times 3^1 = 6

    The LCM of 2 and 6 is 6\boxed{\text{6}}.

Part b

85\frac{8}{5}

بسط (Simplify)

  1. Simplify 85\frac{8}{5}.
  1. Identify common factors between the numerator and the denominator.

    The prime factors of 8 are 2×2×22 \times 2 \times 2.
    The prime factors of 5 are 5.
    There are no common prime factors other than 1.
    The fraction is already in its simplest form.
    The simplified expression is 85\boxed{\frac{8}{5}}.

    1. Simplify nn−2\frac{n}{n-2} with the condition (n<2)(n < 2).
  2. Identify common factors between the numerator and the denominator.

    The numerator is nn. The denominator is (n−2)(n-2).
    These two terms do not share any common factors.
    The expression is already in its simplest form. The condition n<2n < 2 ensures that the denominator n−2≠0n-2 \neq 0.
    The simplified expression is nn−2\boxed{\frac{n}{n-2}}.

    1. Simplify n+2n2−1\frac{n+2}{n^2-1} with the condition (n<1)(n < 1).
  3. Factor the denominator.

    The denominator n2−1n^2-1 is a difference of squares, which factors as (n−1)(n+1)(n-1)(n+1).

    n2−1=(n−1)(n+1)n^2-1 = (n-1)(n+1)
  4. Rewrite the expression with the factored denominator.

    n+2(n−1)(n+1)\frac{n+2}{(n-1)(n+1)}
  5. Identify common factors between the numerator and the denominator.

    The numerator is (n+2)(n+2). The factors in the denominator are (n−1)(n-1) and (n+1)(n+1).
    There are no common factors.
    The expression is already in its simplest form. The condition n<1n < 1 ensures that the denominator (n−1)(n+1)≠0(n-1)(n+1) \neq 0.
    The simplified expression is n+2(n−1)(n+1)\boxed{\frac{n+2}{(n-1)(n+1)}}.

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