ใใใงใฏใๅ้ ๅผใฉใใใฎ็ฉใๆดๅผใฎๅฑ้ใซใคใใฆ่ชฌๆใใพใใ
ๅ้ ๅผใฎ็ฉ
ๆๆฐๆณๅ
ใฏใใใซใๆๅญใฎ็ดฏไนใซใคใใฆ็ขบ่ชใใฆใใใพใใใใ
ๅใๆฐใพใใฏๆๅญใไฝๅใๆใๅใใใใใฎ
็ดฏไนใซใใใฆใๆฐใพใใฏๆๅญใฎๆใๅใใใๅๆฐ
็ดฏไนใซใใใฆใใใ่ฟใๆใใใใๆฐใพใใฏๆๅญ
2ไนใฎๅฅ็งฐ
3ไนใฎๅฅ็งฐ
\(a^n\)ใฏ\(a\)ใ\(n\)ๅๆใๅใใใใใฎใจใใๆๅณใงใใใใใ\(a\)ใฎ\(n\)ไนใจ่จใใพใใใใฎใจใใๅบใฏ\(a\)ใงๆๆฐใฏ\(n\)ใงใใไพใใฐใ$$a^5=a\times a\times a\times a\times a$$ใงใใใ\(a^5\)ใฎๆๆฐใฏ\(5\)ใงใใใพใใ\(a^1=a\)ใงใใใๆๆฐใฎ\(1\)ใฏๆฎ้็็ฅใใใพใใ
ใ\(a\)ใฎๅนณๆนใใฏ\(a^2=a\times a\)ใฎใใจใงใใ\(a\)ใฎ็ซๆนใใฏ\(a^3=a\times a\times a\)ใฎใใจใงใใ
ใๅบใใฏๆฐโ กใง็ปๅ ดใใ่จ่ใงใใใใใฃใใใชใฎใง็ฅใฃใฆใใใพใใใใ
ใใฆใ็ดฏไนใซใคใใฆไปฅไธใฎใๆๆฐๆณๅใใๆใ็ซใกใพใใ
\(m,\,n\)ใๆญฃใฎๆดๆฐใจใใใใใฎใจใใๆฌกใฎ[A]๏ฝ[C]ใๆใ็ซใคใ
$$\begin{align*} [\mathrm{A}]\quad&a^m\cdot a^n=a^{m+n}\\ [\mathrm{B}]\quad&(a^m)^n=a^{mn}\\ [\mathrm{C}]\quad&(ab)^n=a^n b^n\end{align*}$$
ใใใใใฎๅผใฎๆๅณใ่ใใฆใฟใใฐใๆใ็ซใคใใจใฏใใใซๅใใใพใใ
[A]ใซใคใใฆใฏใ\(a^m\)ใฏ\(a\)ใ\(m\)ๅๆใใใใฎใ\(a^n\)ใฏ\(a\)ใ\(n\)ๅๆใใใใฎใชใฎใงใใใใใๆใๅใใใใฐ\(a\)ใ\(m+n\)ๅๆใใใใจใซใชใใพใใใชใฎใงใ\(a^{m+n}\)ใซใชใใพใใ
[B]ใซใคใใฆใฏใ\(a\)ใ\(m\)ๅๆใใใใฎใ\(n\)ๅๆใใใฎใงใๅใใใฆ\(a\)ใฏ\(mn\)ๅใใใพใใ
[C]ใซใคใใฆใฏใ\(a\times b\)ใ\(n\)ๅๆใใใใจใฏใ\(a\)ใจ\(b\)ใๅฅใ ใซใใใใ\(n\)ๅๆใใใใจใจๅใใงใใ
ไปฅไธใซใๆๆฐๆณๅใไฝฟใฃใฆๅ ทไฝ็ใซ่จ็ฎใใไพใ็คบใใพใใฎใงใ็ขบ่ชใใฆใใ ใใใ
(1)\(x^2\cdot x^3\)ใ่จ็ฎใใพใใ
ๆๆฐๆณๅ[A]ใ็จใใใจใ\(x^2\cdot x^3= x^{2+3}=x^5\)ใงใใๅฎ้ใ$$\begin{align*}&\,x^2\cdot x^3\\=&\,(x\cdot x)\cdot(x\cdot x\cdot x)\\=&\,x\cdot x\cdot x\cdot x\cdot x\\=&\,x^5\end{align*}$$ใจใชใใพใใ
(2)\(\qty(x^2)^3\)ใ่จ็ฎใใพใใ
ๆๆฐๆณๅ[B]ใ็จใใใจใ\(\qty(x^2)^3= x^{2\times 3}=x^6\)ใงใใๅฎ้ใ$$\begin{align*}&\,\qty(x^2)^3\\=&\,\qty(x^2)\cdot\qty(x^2)\cdot\qty(x^2)\\=&\,(x\cdot x)\cdot(x\cdot x)\cdot(x\cdot x)\\=&\,x\cdot x\cdot x\cdot x\cdot x\cdot x\\=&\,x^6\end{align*}$$ใจใชใใพใใ
(3)\((xy)^4\)ใ่จ็ฎใใพใใ
ๆๆฐๆณๅ[C]ใ็จใใใจใ\((xy)^4= x^4y^4\)ใงใใๅฎ้ใ$$\begin{align*}&\,(xy)^4\\=&\,(x\cdot y)\cdot(x\cdot y)\cdot(x\cdot y)\cdot (x\cdot y)\\=&\,x\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot y\\=&\,x^4y^4\end{align*}$$ใจใชใใพใใ
ใใฎใใใซใๆๆฐๆณๅใฏ็ดฏไนใฎๆๅณใ่ใใใฐใใใใพใใชๅผใงใใใๆ ฃใใชใใใกใฏ็นใซ[A]ใจ[B]ใซใคใใฆ$$\begin{align*}x^2\times x^3 =& x^{2\times 3}= x^6 \quad \color{red}{\mathrm{NG}}\\ \qty(x^2)^3 =& x^{\qty(2^3)} =x^8\quad\color{red}{\mathrm{NG}}\end{align*}$$ใจใใฆใใพใใในใๅคใใงใใๆณจๆใใฆใใ ใใใ
ๆๆฐๆณๅใฏใใใใใจใใใง็ปๅ ดใใใฎใงใๆฌกใฎ่จ่ใๅ่จ่ใซใใพใใใใ
ๆๆฐใฎๅใฏ็ฉใซๅ่งฃ
ๆๆฐใฎ็ฉใฏในใใฎในใ
$a^{m+n}$ใฏ$a^m$ใจ$a^n$ใฎ็ฉใซๅ่งฃใงใใ๏ผ
$a^{mn}$ใฏ$\qty(a^m)^n$ใพใใฏ$\qty(a^n)^m$ใจๅคๅฝขใงใใ๏ผ
ใๆๆฐใฎๅใฏ ็ฉใซๅ่งฃใใฏๆๆฐๆณๅ[A]ใฎๅผใๅณ่พบใใๅทฆ่พบใฎๅใใซ่ฆใใใฎใงใใใๆๆฐใฎ็ฉใฏ ในใใฎในใใใฏๆๆฐๆณๅ[B]ใฎๅผใๅณ่พบใใๅทฆ่พบใฎๅใใซ่ฆใใใฎใงใใใใใใๅผใ้ๅใใซ่ฆใฆใใพใใใใใกใใฎๅใใฎๅคๅฝขใใใใไฝฟใใพใใฎใงใไธกๆนใฎๅใใงๅผใ่ฆใใใใใใซใชใฃใฆใใใพใใใใ
[ๅ้ ๅผ]ร[ๅ้ ๅผ]
ใงใฏใๆๆฐๆณๅใๆดป็จใใฆๅ้ ๅผใฉใใใฎๆใ็ฎใ่จ็ฎใใพใใใใ
(1)$5xy^2\times \qty(-6x^3y)$ใ่จ็ฎใใพใใ
ๆใ็ฎใฎไบคๆๆณๅใจๆๆฐๆณๅ[A]ใๅฉ็จใใฆใๆฌกใฎใใใซ่จ็ฎใงใใพใใ
$$\begin{align*}&\,5xy^2\times \qty(-6x^3y)\\ =&\,5\times x\times y^2\times (-6)\times x^3\times y\\ =&\,5\times (-6)\times x\times x^3\times y^2\times y\\ =& -30\times x^{1+3}\times y^{2+1}\\ =& -30x^4y^3\end{align*}$$
ใใ ใใๅฎ้ใฎ่จ็ฎใงใฏใใใพใง็ดฐใใ่จ่ฟฐใใๅฟ ่ฆใฏใชใใๆฌกใฎใใใซ้ ญใฎไธญใง่จ็ฎใๆธใพใ็ญใใ ใ่งฃ็ญใใใฐOKใงใใ
ไฟๆฐ๏ผ$5\times (-6)=-30$
$x$ใฎๆๆฐ๏ผ$1+3=4$
$y$ใฎๆๆฐ๏ผ$2+1=3$
ใใใใๆใๅใใใฆใ็ญใใฏ$-30x^4y^3$
(2)$\qty(3x^3y)^2\times \qty(2xy^2)^4$ใ่จ็ฎใใพใใ
ใพใใๆๆฐๆณๅ[C]ใงๅคๅดใฎ็ดฏไนใใฐใใใพใใ
$$\begin{align*}&\,\qty(3x^3y)^2\times \qty(2xy^2)^4\\ =&\, 3^2\cdot \qty(x^3)^2\cdot y^2 \times 2^4\cdot x^4\cdot \qty(y^2)^4\end{align*}$$
ๆฌกใซใๆๆฐๆณๅ[B]ใงใใใใใฎ็ดฏไนใ่จ็ฎใใพใใ
$\mathcal{Core}$โในใใฎในใใฏ ๆๆฐใฎ็ฉ
$$\begin{align*}&\, 3^2\cdot \qty(x^3)^2\cdot y^2 \times 2^4\cdot x^4\cdot \qty(y^2)^4\\ =&\, 9x^{3\times 2}y^2 \times 16x^4y^{2\times 4}\\ =&\, 9x^6y^2\times 16x^4y^8\end{align*}$$
ใใใฆใๆๆฐๆณๅ[A]ใง็ญใใๅบใใพใใ
$$\begin{align*}&\, 9x^6y^2\times 16x^4y^8\\ =&\, 9\cdot 16\cdot x^{6+4}y^{2+8}\\ =&\, 144x^{10}y^{10}\end{align*}$$
ใใใ่ณๅ ใง่จ็ฎใงใใใใใซใชใใพใใใใ
ไฟๆฐ๏ผ$3^2\times 2^4=9\times 16 \overset{\color{blue}{\spadesuit}}{=}160-16=144$♠9ๅใฏ10ๅใใฆๅผใ๏ผ
$x$ใฎๆๆฐ๏ผ$3\times 2+4=10$
$y$ใฎๆๆฐ๏ผ$2+2\times 4=10$
ใใใใๆใๅใใใฆใ็ญใใฏ$144x^{10}y^{10}$
็ทด็ฟๅ้กโ
ๅ้ ๅผใฉใใใฎ็ฉใซใคใใฆ็ทด็ฟใใพใใใใๆ็ฎใงในใใผใใฃใผใซ่จ็ฎใใใใจใๆ่ญใใฆใใ ใใใ
ๅ้ก
ๆฌกใฎ่จ็ฎใใใใ
(1)$2a^4b^3\times 5a^3b^2$
(2)$\qty(-3x^3y)^2\times \qty(-4x^2y^3)$
(3)$\qty(a^3b^2c)^3\times \qty(5a^2b^4c^3)^2$
่งฃ่ชฌ
(1)
ไฟๆฐ๏ผ$2\times 5=10$
$a$ใฎๆๆฐ๏ผ$4+3=7$
$b$ใฎๆๆฐ๏ผ$3+2=5$
(2)
ไฟๆฐ๏ผ$(-3)^2\times (-4)=9\times (-4)=-36$
$x$ใฎๆๆฐ๏ผ$3\times 2+2=8$
$y$ใฎๆๆฐ๏ผ$1\times 2+3=5$
(3)
ไฟๆฐ๏ผ$1\times 5^2=25$
$a$ใฎๆๆฐ๏ผ$3\times 3+2\times 2=13$
$b$ใฎๆๆฐ๏ผ$2\times 3+4\times 2=14$
$c$ใฎๆๆฐ๏ผ$1\times 3+3\times 2=9$
่งฃ็ญ
(1)$10a^7b^5$
(2)$-36x^8y^5$
(3)$25a^{13}b^{14}c^9$
ๆดๅผใฎๅฑ้
[ๅ้ ๅผ]ร[ๅค้ ๅผ]
็ถใใฆใๅ้ ๅผใจๅค้ ๅผใฎๆใ็ฎใ่ฆใฆใใใพใใใใ
(1)$4x(3x+5y)$ใ่จ็ฎใใพใใ
ๅ้ ๆณๅใ็จใใฆใใซใใณใ้ใใพใใใใ
$$4x(3x+5y)=4x\cdot 3x+4x\cdot 5y$$
$+$ใฎๅๅพใๅ้ ๅผใฉใใใฎ็ฉใจใชใใฎใงใๅ ใซ่ชฌๆใใ่ฆ้ ใง่จ็ฎใใพใใ
$$4x\cdot 3x+4x\cdot 5y=12x^2+20xy$$
(2)$\qty(6x^2+4xy+3y^2)\qty(-xy^3)^2$ใ่จ็ฎใใพใใ
็ดฏไนใฎ่จ็ฎๅชๅ ๅบฆใฏไปใฎๆใ็ฎใๅฒใ็ฎใใใ้ซใใงใใใงใใฎใงใๅ ใซๅพใๅดใฎ็ดฏไนใ่จ็ฎใใพใใใใ
$$\qty(6x^2+4xy+3y^2)\qty(-xy^3)^2=(6x^2+4xy+3y^2)(x^2y^6)$$
ๅ้ ๅผใๅพใใงใใฃใฆใใๅใใใใซๅ้ ๆณๅใไฝฟใใพใใ
$$\begin{align*}&(6x^2+4xy+3y^2)(x^2y^6)\\=&\,6x^2\cdot x^2y^6+4xy\cdot x^2y^6+3y^2\cdot x^2y^6\\ =&\,6x^4y^6+4x^3y^7+3x^2y^8\end{align*}$$
ๅค้ ๅผใซๅฅใฎๆดๅผใๆใใใจใใฏใๅ้ ๆณๅใๅฉ็จใใฆใซใใณใๅคใใพใใใใฎใใใชๆไฝใ็ตใฆๆดๅผใฎ็ฉใ่จ็ฎใใใใจใใๅฑ้ใใจ่จใใพใใ
่คๆฐใฎๆดๅผ(ใใกๅฐใชใใจใ1ใคใฏๅค้ ๅผ)ใฎ็ฉใ1ใคใฎๆดๅผใง่กจ็พใใใใจ
[ๅค้ ๅผ]ร[ๅค้ ๅผ]
ๅค้ ๅผใฉใใใฎ็ฉ(ๅฑ้)ใๅ้ ๆณๅใซใใฃใฆ่จ็ฎใงใใพใใ
(1)$(2x+3y)(4x-5y)$ใๅฑ้ใใพใใ
ใพใใๅพใใฎใซใใณใฎไธญใฎๅผใใฒใจใใใพใใจ่ฆใฆใๅ้ ๆณๅใๅฉ็จใใฆๅใฎใซใใณใๅคใใพใใ
$$(2x+3y)\enclose{box}[mathbackground=cyan]{(4x-5y)}=2x\,\enclose{box}[mathbackground=cyan]{(4x-5y)}+3y\,\enclose{box}[mathbackground=cyan]{(4x-5y)}$$
ใใใซใใใใใใๅ้ ๆณๅใงใใฉใใพใใๅ้ก้ ใฏใพใจใใฆใใใพใใใใ
$$\begin{align*}&\,2x(4x-5y)+3y(4x-5y)\\=&\,2x\cdot 4x+2x\cdot (-5y)+3y\cdot 4x+3y\cdot (-5y)\\=&\,8x^2-10xy+12xy-15y^2\\=&\,8x^2+2xy-15y^2\end{align*}$$
ใใๅฐใใใฐใใ่จ็ฎใใฆใฟใพใใใใๅใใใใใใ$(A+B)(C+D)$ใจใใๅผใๅใใใใซๅฑ้ใใฆใฟใพใใ
$$\begin{align*}&(\color{red}{A}\color{blue}{+B})(\color{green}{C}\color{violet}{+D})\\=&\,\color{red}{A}(\color{green}{C}\color{violet}{+D})+\color{blue}{B}(\color{green}{C}\color{violet}{+D})\\=&\,\color{red}{A}\color{green}{C}+\color{red}{A}\color{violet}{D}+\color{blue}{B}\color{green}{C}+\color{blue}{B}\color{violet}{D}\end{align*}$$
ๅฑ้ๅพใฎๅผใใใ่ฆใใจใใใใใใฎใซใใณใฎไธญใใ้ ใ1ใคใใค้ธใใงๆใๅใใใใใจใใในใฆใฎ็ตใฟๅใใใง่กใฃใฆใใใใใ่ถณใๅใใใใใจใง็ญใใฎๅผใๅพใใใใใจใใใใใงใใใใใใใฎใใใซ่กใใฐใใใกใใกๅ้ ๆณๅใไฝฟใฃใฆใซใใณใ1ใคใใคใฐใใๅฟ ่ฆใใชใใชใใพใใๅ้กใฎๅผใๅใ่ฆ้ ใง่จ็ฎใใฆใฟใพใใใใ
$$\begin{align*}&(\color{red}{2x}\color{blue}{+3y})(\color{green}{4x}\color{violet}{-5y})\\=&\,\color{red}{2x}\cdot\color{green}{4x}+\color{red}{2x}\cdot\color{violet}{(-5y)}+\color{blue}{3y}\cdot\color{green}{4x}+\color{blue}{3y}\cdot\color{violet}{(-5y)}\\=&\,8x^2-10xy+12xy-15y^2\\=&\,8x^2+2xy-15y^2\end{align*}$$
(2)$(x+3y-5)\qty(x^2-3xy+4y^2)$ใๅฑ้ใใพใใ
ๅค้ ๅผใฎ้ ใๅขใใฆใ่ฆ้ ใฏๅใใงใใใใใใใฎๅค้ ๅผใใ้ ใ1ใคใใค้ธใใงๆใๅใใใใใจใใใใชใใในใฆใฎ็ตใฟๅใใใง่กใใพใใใใ
$$\begin{align*}&(x+3y-5)\qty(x^2-3xy+4y^2)\\=&\,x\cdot x^2+x\cdot(-3xy)+x\cdot 4y^2+3y\cdot x^2+3y\cdot(-3xy)+3y\cdot 4y^2+(-5)\cdot x^2+(-5)\cdot(-3xy)+(-5)\cdot 4y^2\\=&\,x^3-3x^2y+4xy^2+3x^2y-9xy^2+12y^3-5x^2+15xy-20y^2\\=&\,x^3-5xy^2+12y^3-5x^2+15xy-20y^2\end{align*}$$
ๆ ฃใใฆใใใใ้ไธญๅผใฏใฉใใฉใ็็ฅใใฆ่จ็ฎใงใใใใใซใใฆใใใพใใใใ
็ทด็ฟๅ้กโก
ๅฑ้ใฎ็ทด็ฟใใใพใใใใในใใผใใๆ่ญใใคใคใ่จ็ฎใในใ็ตถๅฏพใซ่ตทใใใชใใใใซๆณจๆใใฆ่จ็ฎใใฆใใใพใใใใๆ ฃใใฆใใใใ้ไธญๅผใ1่กใใค็็ฅใใฆ่จ็ฎใงใใใใใซใใพใใใใ
ๅ้ก
ๆฌกใฎๅผใๅฑ้ใใใ
(1)$3a^2b\qty(5a+6b^2)$
(2)$\qty(4x^2+xy-2y^2)\qty(3x^2y)^3$
(3)$\qty(x^2+4)(x-y)$
(4)$(-a+3b-2c)(4a-2b+3c)$
(5)$\qty(4-3t+t^3)\qty(-t+2t^2+5)$
(6)$(x+1)(y+2)(z+3)$
่งฃ่ชฌ
$$\begin{align*}(1)\quad&\,3a^2b\qty(5a+6b^2)\\=&\,3a^2b\cdot 5a+3a^2b\cdot 6b^2\\=&\,15a^3b+18a^2b^3\end{align*}$$
$$\begin{align*}(2)\quad&\qty(4x^2+xy-2y^2)\qty(3x^2y)^3\\=&\qty(4x^2+xy-2y^2)\qty(27x^6y^3)\\=&\,4x^2\cdot 27x^6y^3+xy\cdot 27x^6y^3+\qty(-2y^2)\cdot 27x^6y^3\\=&\,108x^8y^3+27x^7y^4-54x^6y^5\end{align*}$$
$$\begin{align*}(3)\quad&\qty(x^2+4)(x-y)\\=&\,x^2\cdot x+x^2\cdot(-y)+4\cdot x+4\cdot(-y)\\=&\,x^3-x^2y+4x-4y\end{align*}$$
$$\begin{align*}(4)\quad&(-a+3b-2c)(4a-2b+3c)\\=&(-a)\cdot 4a+(-a)\cdot(-2b)+(-a)\cdot 3c+3b\cdot 4a+3b\cdot (-2b)+3b\cdot 3c+(-2c)\cdot 4a+(-2c)\cdot(-2b)+(-2c)\cdot 3c\\=&\,-4a^2+2ab-3ac+12ab-6b^2+9bc-8ac+4bc-6c^2\\=&\,-4a^2-6b^2-6c^2+14ab+13bc-11ca\end{align*}$$
$ab\rightarrow bc\rightarrow ca$ใจใใๆด็ใฎใใใใ่ผช็ฐใฎ้ ใจ่จใใพใใๆๅญใ3็จฎ้กใฎๆๅญๅผใๆด็ใใใจใใใใฎ้ ใซๆด็ใใใจใใใใใใใชใฃใฆใใใใใงใใ
$$\begin{align*}(5)\quad&\qty(4-3t+t^3)\qty(-t+2t^2+5)\\=&\,4\cdot(-t)+4\cdot 2t^2+4\cdot 5+(-3t)\cdot(-t)+(-3t)\cdot 2t^2+(-3t)\cdot 5+t^3\cdot(-t)+t^3\cdot 2t^2+t^3\cdot 5\\=&\,-4t+8t^2+20+3t^2-6t^3-15t-t^4+2t^5+5t^3\\=&\,2t^5-t^4-t^3+11t^2-19t+20\end{align*}$$
็ญใใฏ้ในใใฎ้ ใซๆด็ใใฆๆธใ็ฟๆ ฃใใคใใพใใใใ
(6)ๅ ใซ2ใคใฎใซใใณใใฐใใๆนๆณใจใ3ใคใฎใซใใณใใพใจใใฆๅฑ้ใใๆนๆณใใใใพใใ
2ใคใฎใซใใณใใฐใใๆนๆณใซใคใใฆใไพใใฐ1็ช็ฎใจ2็ช็ฎใฎใซใใณใๅ ใซๅฑ้ใใใจๆฌกใฎใใใช่จ็ฎใซใชใใพใใ
$$\begin{align*} \quad&(x+1)(y+2)(z+3)\\=&(x\cdot y+x\cdot 2+1\cdot y+1\cdot 2)(z+3)\\=&(xy+2x+y+2)(z+3)\\=&\,xy\cdot z+xy\cdot 3+2x\cdot z+2x\cdot 3+y\cdot z+y\cdot 3+2\cdot z+2\cdot 3\\=&\,xyz+3xy+2xz+6x+yz+3y+2z+6 \;\color{magenta}{\leftarrow ใใใพใงใงใใใ}\\=&\,xyz+3xy+yz+2zx+6x+3y+2z+6\;\color{magenta}{\leftarrow ใใใพใงๆด็ใงใใใจ\mathrm{GREAT}}\end{align*}$$
3ใคใฎใซใใณใใพใจใใฆๅฑ้ใใใจใๆฌกใฎใใใซใชใใพใใ3ใคใฎใซใใณใๅฑ้ใใใจใใใใใใใฎใซใใณใใ้ ใ1ใคใใค้ธใใงๆใๅใใใใใจใฏๅคใใใพใใใใซใใณใๅคใใถใ็ตใฟๅใใใๅคใใชใใฎใงใๆผใใซๆณจๆใใฆใใ ใใใ
$$\begin{align*}&(x+1)(y+2)(z+3)\\=&\,x\cdot y\cdot z+x\cdot y\cdot 3+x\cdot 2\cdot z+x\cdot 2\cdot 3+1\cdot y\cdot z+1\cdot y\cdot 3+1\cdot 2\cdot z+1\cdot 2\cdot 3\\=&\,xyz+3xy+2xz+6x+yz+3y+2z+6\\=&\,xyz+3xy+yz+2zx+6x+3y+2z+6\end{align*}$$
่งฃ็ญ
(1)$15a^3b+18a^2b^3$
(2)$108x^8y^3+27x^7y^4-54x^6y^5$
(3)$x^3-x^2y+4x-4y$
(4)$-4a^2-6b^2-6c^2+14ab+13bc-11ca$
(5)$2t^5-t^4-t^3+11t^2-19t+20$
(6)$xyz+3xy+yz+2zx+6x+3y+2z+6$
่ฃ่ถณ
ๆๆฐใฎๆกๅผต
โปใใฎ้ ็ฎใซใฏๆฐๅญฆโ กใฎๅญฆ็ฟๅ ๅฎนใๅซใพใใพใใ
ใใฎใใผใธใงใฏ่ช็ถๆฐไนใฎ็ดฏไนใ ใใ่ใใพใใใ็ดฏไนใฎๅฎ็พฉใใๅใใใฎใไฝๅใๆใๅใใใใใจใใฆใใใใใฎใไฝๅใใใจใใใฎใฏ่ช็ถๆฐๅๆใใใใจใใๆณๅฎใงใใชใใใใงใใใ$3$ใ$2.5$ๅๆใๅใใใใใจใใใฎใฏไธ่ช็ถใงใใใญใใ ใใใๆๆฐใฏ่ช็ถๆฐใใ่ใใใใจใใงใใชใใฃใใฎใงใใใใใฆใ่ช็ถๆฐใจใใ้ใใใๆๆฐใฎไธญใงๆๆฐๆณๅใๆ็ซใใใใจใ็ขบ่ชใใพใใใ
ใใใใๆฐๅญฆ่ ใฏใๆๆฐใซ่ช็ถๆฐใใๅ ฅใใใใชใใชใใฆ็ชฎๅฑใใใชใใใใใฃใจใใใใชๆฐใๅ ฅใใใใใใใซใใใใใจๆใใพใใๆฐๅญฆใจใใๅญฆๅใฏใใฎใใใซใใฆๅบใใฃใฆใใฃใฆใใใใใงใใๆฐๅญฆ่ ใฏๆๆฐใซ$0$ใ่ฒ ใฎๆฐใๅๆฐใๅฐๆฐใชใฉใๅ ฅใใใใใใใซ่ใใพใใใใใฎ้ใซใๆๆฐใ่ช็ถๆฐใฎใจใใซๆใ็ซใฃใฆใใๆๆฐๆณๅใๆใ็ซใคใใใซใใใใจ่ใใใฎใงใใใใฎ็ตๆใๆฌกใฎใใใซๅฎใใใจๆๆฐๆณๅใๆใ็ซใคใใใซๆๆฐใ่ช็ถๆฐไปฅๅคใซใงใใใใจใ็บ่ฆใใพใใใ
$$\begin{align*}a^0=&1\quad(a\neq 0)\\a^{-b}=&\frac{1}{\,a^b\,}\quad(a>0)\\a^{\frac{m}{n}}=&\sqrt[n]{a^m\,}\quad(a>0,\,m\,ใฏๆดๆฐ,\,n\,ใฏ่ช็ถๆฐ)\end{align*}$$
$\sqrt[n]{a^m\,}$ใฏใ$n$ไนใใใจ$a^m$ใซใชใๆฐใใจใใๆๅณใงใใๆๆฐใ่ช็ถๆฐใงใชใ็ดฏไนใซใคใใฆใฏๆฐโ กใง่ฉณใใๅญฆ็ฟใใพใใใใใฎใใใซๅฎใใใจๆๆฐๆณๅใๆใ็ซใคใใจใ็ขบ่ชใใฆใฟใฆใใ ใใใ
ใพใจใ
- ๅ้ ๅผใฉใใใฎๆใ็ฎใฏๆๆฐๆณๅใๅฉ็จใใ
- ๅผใฎๅฑ้ใฏใใใใใฎใซใใณใฎไธญใใ้ ใ1ใคใใค้ธใใงๆใๅใใใใใจใใในใฆใฎ็ตใฟๅใใใง่กใฃใฆ่ถณใๅใใใ
- ่จ็ฎใในใซๆณจๆใใชใใ้ๅบฆใขใใใฎ็ทด็ฟใใใใ