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Find the value of (m⁵)³ n² + n⁶ m / nm⁴ - m⁷, if n = 4 and m = 1.

Mathematics

ScanSolve AI Answer

Answer

21

72\dfrac{7}{2}

60815\dfrac{608}{15}

Step-by-step solution

10 steps

  1. Recall the difference of cubes formula

    n3−m3=(n−m)(n2+nm+m2).n^3 - m^3 = (n - m)(n^2 + nm + m^2).

    Thus,

    n3−m3n−m=n2+nm+m2.\frac{n^3 - m^3}{n - m} = n^2 + nm + m^2.

  2. Substitute n=4n = 4 and m=1m = 1

    n2=42=16,n^2 = 4^2 = 16, nm=4⋅1=4,nm = 4 \cdot 1 = 4, m2=12=1.m^2 = 1^2 = 1.

  3. Add the terms

    16+4+1=21.16 + 4 + 1 = 21.

    21

  4. Compute r−sr - s

    r−s=3−1=2.r - s = 3 - 1 = 2.

  5. Compute t−st - s

    t−s=2−1=1.t - s = 2 - 1 = 1.

  6. Compute the numerator (r−s)3−(t−s)2(r - s)^3 - (t - s)^2

    (r−s)3=23=8,(r - s)^3 = 2^3 = 8, (t−s)2=12=1,(t - s)^2 = 1^2 = 1, 8−1=7.8 - 1 = 7.

  7. Divide by the denominator r−sr - s

    72.\frac{7}{2}.

    \dfrac{7}{2}

  8. Factor the expression algebraically

    (ab)3+(bc)3=a3b3+b3c3=b3(a3+c3),(ab)^3 + (bc)^3 = a^3 b^3 + b^3 c^3 = b^3 (a^3 + c^3), (ab)3+(bc)3abc=b3(a3+c3)abc=b2(a3+c3)ac.\frac{(ab)^3 + (bc)^3}{abc} = \frac{b^3 (a^3 + c^3)}{abc} = \frac{b^2 (a^3 + c^3)}{ac}.

  9. Substitute a=5a=5, b=2b=2, c=3c=3

    a3=53=125,a^3 = 5^3 = 125, c3=33=27,c^3 = 3^3 = 27, a3+c3=125+27=152,a^3 + c^3 = 125 + 27 = 152, b2=22=4,b^2 = 2^2 = 4, ac=5⋅3=15.ac = 5 \cdot 3 = 15.

  10. Multiply and divide

    4⋅152=608,4 \cdot 152 = 608, 60815.\frac{608}{15}.

    \dfrac{608}{15}

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