Chapter 49, Problem 8AR

To determine

(a)

Solve the expression ${(-4a^{3}b{c}^2)}^3$

Answer to Problem 8AR

  $Onsolvingthe\exp ression{(-4a^{3}b{c}^2)}^{3}weget64a^9{b}^3{c}^6$

Explanation of Solution

Concept Used:

  ${(-1)}^n=1,whenniseven$

  ${(-1)}^n=-1,whennisodd$

Calculation:

  ${(-4a^{3}b{c}^2)}^3=-1^3\times 4^3{\left(a^3\right)}^3{b}^3{\left(c^2\right)}^3$

  ${(-4a^{3}b{c}^2)}^3=-64a^9{b}^3{c}^6$

Conclusion:

  $Onsolvingthe\exp ression{(-4a^{3}b{c}^2)}^{3}weget64a^9{b}^3{c}^6$

To determine

(b)

Solve the expression ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2$

Answer to Problem 8AR

  $Onsolvingthe\exp ression{[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^{2}weget{x}^6{y}^{12}+x^4{y}^4-2x^5{y}^8$

Explanation of Solution

Concept used:

  ${(x+y)}^2=x^2+2xy+y^2$

Calculation:

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2={[x^3{\left(y^2\right)}^3-\left(x^2{y}^2\right)]}^2$

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2={[x^3{y}^6-x^2{y}^2]}^2$

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2=x^4{y}^4{[x{y}^4-1]}^2$

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2=x^4{y}^4[{\left(x{y}^4\right)}^2+1-2\times x{y}^4]$

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2=x^4{y}^4[x^2{y}^8+1-2\times x{y}^4]$

  ${[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^2=x^6{y}^{12}+x^4{y}^4-2x^5{y}^8$

Conclusion:

  $Onsolvingthe\exp ression{[{\left(x{y}^2\right)}^3-\left(x^2{y}^2\right)]}^{2}weget{x}^6{y}^{12}+x^4{y}^4-2x^5{y}^8$

To determine

(c)

Solve the expression ${(-27a^6{b}^3{c}^9)}^{1/3}$

Answer to Problem 8AR

  $Onsolvingthe\exp ression{(-27a^6{b}^3{c}^9)}^{1/3}weget-3a^{2}b{c}^3$

Explanation of Solution

Concept Used:

  ${(-1)}^{1/3}=-1$

Calculation:

  ${(-27a^6{b}^3{c}^9)}^{1/3}=-1^{1/3}\times {27}^{1/3}{\left(a^6\right)}^{1/3}{b}^{3/3}{\left(c^9\right)}^{1/3}$

  ${(-27a^6{b}^3{c}^9)}^{1/3}=-3a^{2}b{c}^3$

Conclusion:

  $Onsolvingthe\exp ression{(-27a^6{b}^3{c}^9)}^{1/3}weget-3a^{2}b{c}^3$

To determine

(d)

Solve the expression $(6x^2-y^3)(-3x^3+y^2)$

Answer to Problem 8AR

  $Onsolvingthe\exp ression(6x^2-y^3)(-3x^3+y^2)weget-18x^5+6x^2{y}^2+3x^3{y}^3-y^5$

Explanation of Solution

Calculation:

  $(6x^2-y^3)(-3x^3+y^2)=6x^2\times \left(-3x^3\right)+6x^2\times y^2-y^3\times \left(-3x^3\right)-y^3{y}^2$

  $(6x^2-y^3)(-3x^3+y^2)=-18x^5+6x^2{y}^2+3x^3{y}^3-y^5$

Conclusion:

  $Onsolvingthe\exp ression(6x^2-y^3)(-3x^3+y^2)weget-18x^5+6x^2{y}^2+3x^3{y}^3-y^5$

To determine

(e)

Solve the expression $-6(x^2+y-2x)$

Answer to Problem 8AR

  $Onsolvingthe\exp ression-6(x^2+y-2x)weget-6x^2-6y+12x$

Explanation of Solution

Calculation:

  $\begin{array}{l}-6(x^2+y-2x)=-6x^2-6y+12x\\ \end{array}$

Conclusion:

  $Onsolvingthe\exp ression-6(x^2+y-2x)weget-6x^2-6y+12x$

To determine

(f)

Solve the expression $36-{(m^3+m)}^3+(m^3+12)$

Answer to Problem 8AR

  $Onsolvingthe\exp ression36-{(m^3+m)}^3+(m^3+12)weget36-m^9-3m^7-3m^5+12$

Explanation of Solution

Concept Used:

  ${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

Calculation:

  $36-{(m^3+m)}^3+(m^3+12)=36-(m^9+3{\left(m^3\right)}^{2}m+3m^3{m}^2+m^3)+(m^3+12)$

  $36-{(m^3+m)}^3+(m^3+12)=36-m^9-3m^7-3m^5-m^3+m^3+12$

  $36-{(m^3+m)}^3+(m^3+12)=36-m^9-3m^7-3m^5+12$

Conclusion:

  $Onsolvingthe\exp ression36-{(m^3+m)}^3+(m^3+12)weget36-m^9-3m^7-3m^5+12$

To determine

(g)

Solve the expression $-3a{(2a)}^2+\sqrt{36a^4}$

Answer to Problem 8AR

  $Onsolvingthe\exp ression-3a{(2a)}^2+\sqrt{36a^4}weget-6a^2(2a-1)$

Explanation of Solution

Concept Used:

  $\sqrt{a^2}=+aor-a$

Calculation:

  $\sqrt{36a^4}=6a^{2}or-6(-a^2)$

  $\sqrt{36a^4}=6a^2$

  $-3a{(2a)}^2+\sqrt{36a^4}=-12a^3+6a^2$

  $-3a{(2a)}^2+\sqrt{36a^4}=-6a^2(2a-1)$

Conclusion:

  $Onsolvingthe\exp ression-3a{(2a)}^2+\sqrt{36a^4}weget-6a^2(2a-1)$

To determine

(h)

Solve the expression $9y-x[-8+{(xy)}^2-y]+12x$

Answer to Problem 8AR

  $Onsolvingthe\exp ression9y-x[-8+{(xy)}^2-y]+12xweget-x^3{y}^2+xy+20x+9y$

Explanation of Solution

Calculation:

  $9y-x[-8+{(xy)}^2-y]+12x=9y+8x-x^3{y}^2+xy+12x$

  $9y-x[-8+{(xy)}^2-y]+12x=-x^3{y}^2+xy+20x+9y$

Conclusion:

  $Onsolvingthe\exp ression9y-x[-8+{(xy)}^2-y]+12xweget-x^3{y}^2+xy+20x+9y$

To determine

(i)

Solve the expression $b{(b+7m)}^2-b{(b-7m)}^2$

Answer to Problem 8AR

  $Onsolvingthe\exp ressionb{(b+7m)}^2-b{(b-7m)}^{2}weget28m{b}^2$

Explanation of Solution

Concept used:

  ${(x+y)}^2=x^2+2xy+y^2$

Calculation:

  $b{(b+7m)}^2-b{(b-7m)}^2=b(b^2+2\times 7m\times b+{(7m)}^2)-b(b^2-2\times 7m\times b+{(7m)}^2)$

  $b{(b+7m)}^2-b{(b-7m)}^2=b(b^2+2\times 7m\times b+49m^2-(b^2-2\times 7m\times b+49m^2))$

  $b{(b+7m)}^2-b{(b-7m)}^2=28m{b}^2$

Conclusion:

  $Onsolvingthe\exp ressionb{(b+7m)}^2-b{(b-7m)}^2,weget28m{b}^2$