Question

Simplify these thing below please. I am stuck again... Thank you

Answer 1

`\textsf{1.} \quad 18\;x^3\;y^(6)√(14x)`

`\textsf{2.} \quad -8√(2)`

Explanation:

Question 1

Given expression:

`9\sqrt{56x^7y^(12)}`

`\textsf{Apply radical rule} \quad √(ab)=√(a)√(b):`

`\implies 9√(56)√(x^7)\sqrt{y^(12)}`

Rewrite 56 as 4·14:

`\implies 9√(4 \cdot 14)√(x^7)\sqrt{y^(12)}`

`\textsf{Apply radical rule} \quad √(ab)=√(a)√(b):`

`\implies 9√(4)√(14)√(x^7)\sqrt{y^(12)}`

Rewrite 4 as 2²:

`\implies 9√(2^2)√(14)√(x^7)\sqrt{y^(12)}`

Simplify:

`\implies 9\cdot 2√(14)√(x^7)\sqrt{y^(12)}`

`\implies 18√(14)√(x^7)\sqrt{y^(12)}`

`\textsf{Apply exponent rule} \quad √(a)=a^{(1)/(2)}:`

`\implies 18√(14)\;(x^7)^{(1)/(2)}\;(y^(12))^{(1)/(2)}`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies 18√(14)\;x^{(7)/(2)}\;y^{(12)/(2)}`

`\implies 18√(14)\;x^{(7)/(2)}\;y^6`

Rewrite ⁷/₂ as 3 + ¹/₂

`\implies 18√(14)\;x^{(3+(1)/(2))}\;y^(6)`

`\textsf{Apply exponent rule} \quad a^(b+c)= a^b \cdot a^c:`

`\implies 18√(14)\;x^3 \; x^{(1)/(2)}\;y^(6)`

`\textsf{Apply exponent rule} \quad a^{(1)/(2)}=√(a):`

`\implies 18√(14)\;x^3 \; √(x)\;y^(6)`

Rearrange:

`\implies 18\;x^3\;y^(6)√(14x)`

Question 2

Given expression:

`7√(32)-6√(72)`

Rewrite 32 as 16·2 and 72 as 36·2:

`\implies 7√(16 \cdot 2)-6√(36 \cdot 2)`

`\textsf{Apply radical rule} \quad √(ab)=√(a)√(b):`

`\implies 7√(16)√(2)-6√(36)√(2)`

Rewrite 16 as 4² and 36 as 6²:

`\implies 7√(4^2)√(2)-6√(6^2)√(2)`

`\textsf{Apply radical rule} \quad √(a^2)=a, \quad a \geq 0:`

`\implies 7 \cdot 4√(2)-6\cdot 6√(2)`

Simplify:

`\implies 28√(2)-36√(2)`

`\implies -8√(2)`

Answer 2

`9\sqrt{56 x^7 y^(12)}\qquad \begin{cases} 56=7\cdot 2\cdot 2\cdot 2\\ \qquad 7\cdot 2^2 \cdot 2\\ \qquad 2^2\cdot 14\\ x^7=x^((3)(2)+1)\\ \qquad (x^3)^2\cdot x^1\\ y^(12)=y^((6)(2))\\ \qquad (y^6)^2 \end{cases}\hspace{5em} \begin{array}{llll} 9√(2^2(14)(x^3)^2 x (y^6)^2) \\\\\\ 9(2)(x^3)(y^6)√(14x) \\\\\\ {\Large \begin{array}{llll} 18x^3y^6√(14x) \end{array}} \end{array}`