Answer:
[tex]\textsf{1.}\quad \sf 18x^{12}y^4z^8 =H[/tex]
[tex]\textsf{2.}\quad \sf 64x^{12}y^3z^9=Y[/tex]
[tex]\textsf{3.}\quad \sf 30x^3y^2 =I[/tex]
[tex]\textsf{4.}\quad \sf 4xz=S[/tex]
[tex]\textsf{5.}\quad \sf \dfrac{-3y^4}{x^6z^{11}}=F[/tex]
[tex]\textsf{6.}\quad \sf 6x^4y^4=R[/tex]
[tex]\textsf{7.}\quad \sf 20 x^{10}y^5z^{10}=C[/tex]
[tex]\textsf{8.}\quad \sf 72x^8z^4=E[/tex]
Mystery phrase: CHEESY FRIES
Step-by-step explanation:
To match each problem to its equivalent simplified expression, we can use the laws of exponents:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Laws of Exponents}}\\\\\textsf{Product:}\;\;a^m \times a^n=a^{m+n}\\\\\textsf{Quotient:}\;\;a^m \div a^n=a^{m-n}\\\\\textsf{Power of a Power:}\;\;(a^m)^n=a^{mn}\\\\\textsf{Power of a Quotient:}\;\;\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\\textsf{Power of a Product:}\;\;(ab)^m=a^mb^m\\\\\textsf{Negative Exponent:}\;\;a^{-m}=\dfrac{1}{a^m}\\\\\textsf{Zero Exponent:}\;\;a^0=1\end{array}}[/tex]
Question 1
[tex]\sf \left(3x^5yz^7\right)\left(6x^7y^3z\right)\\\\\\3 \cdot 6\cdot x^5\cdot x^7\cdot y\cdot y^3\cdot z^7\cdot z\\\\\\18\cdot x^{5+7}\cdot y^{1+3}\cdot z^{7+1}\\\\\\18x^{12}y^4z^8 \rightarrow \boxed{\sf H}[/tex]
Question 2
[tex]\sf \left(4x^4yz^3\right)^3\\\\\\4^3\cdot (x^4)^3\cdot y^3\cdot (z^3)^3\\\\\\64\cdot x^{4\cdot 3}\cdot y^3\cdot z^{3 \cdot 3}\\\\\\64x^{12}y^3z^9=\boxed{\sf Y}[/tex]
Question 3
[tex]\sf \left(-5x^{-2}\right)\left(-2y\right)\left(3x^5y\right)\\\\\\-5 \cdot -2 \cdot 3 \cdot x^{-2}\cdot x^5\cdot y\cdot y\\\\\\30\cdot x^{-2+5}\cdot y^{1+1}\\\\\\30x^3y^2 =\boxed{\sf I}[/tex]
Question 4
[tex]\sf \left(16x^2y^0z^2\right)^{\frac12}\\\\\\16^{\frac12}\cdot (x^2)^{\frac12}\cdot (y^0)^{\frac12}\cdot (z^2)^{\frac12}\\\\\\4\cdot x^{2 \cdot \frac12}\cdot 1^{\frac12}\cdot z^{2 \cdot \frac12}\\\\\\4\cdot x^1\cdot 1 \cdot z^1\\\\\\4xz=\boxed{\sf S}[/tex]
Question 5
[tex]\sf \dfrac{-27x^{-2}y^4z^6}{9x^4z^{17}}\\\\\\\dfrac{-27}{9}\cdot \dfrac{x^{-2}}{x^4}\cdot y^4\cdot \dfrac{z^6}{z^{17}}\\\\\\-3\cdot x^{-2-4}\cdot y^4\cdot z^{6-17}\\\\\\-3 \cdot x^{-6}\cdot y^4\cdot z^{-11}\\\\\\\dfrac{-3y^4}{x^6z^{11}}=\boxed{\sf F}[/tex]
Question 6
[tex]\sf x^2y^3\left(6x^2y\right)\\\\\\6 \cdot x^2 \cdot x^2 \cdot y^3 \cdot y\\\\\\6\cdot x^{2+2}\cdot y^{3+1}\\\\\\6x^4y^4=\boxed{\sf R}[/tex]
Question 7
[tex]\sf \dfrac{\left(5y^3z^{13}\right)\left(12x^4y^2z^4\right)}{3x^{-6}z^7}\\\\\\\dfrac{5\cdot 12 \cdot x^4 \cdot y^3 \cdot y^2 \cdot z^{13}\cdot z^4}{3x^{-6}z^7}\\\\\\\dfrac{60 \cdot x^4 \cdot y^{3+2} \cdot z^{13+4}}{3x^{-6}z^7}\\\\\\\dfrac{60 \cdot x^4 \cdot y^{5} \cdot z^{17}}{3x^{-6}z^7}\\\\\\\dfrac{60}{3} \cdot \dfrac{x^4}{x^{-6}} \cdot y^5\cdot \dfrac{z^{17}}{z^7} \\\\\\20 \cdot x^{4-(-6)} \cdot y^5 \cdot z^{17-7}\\\\\\20 x^{10}y^5z^{10}=\boxed{\sf C}[/tex]
Question 8
[tex]\sf \left(6x^3y^0\right)^2 \left(2x^2z^4\right)\\\\\\6^2 \cdot (x^3)^2\cdot (y^0)^2\cdot 2\cdot x^2\cdot z^4\\\\\\36\cdot x^{3\cdot 2}\cdot 1^2\cdot 2\cdot x^2\cdot z^4\\\\\\36\cdot x^6\cdot 1\cdot 2\cdot x^2\cdot z^4\\\\\\36\cdot 1\cdot 2\cdot x^6\cdot x^2\cdot z^4\\\\\\72\cdot x^{6+2}\cdot z^4\\\\\\72x^8z^4=\boxed{\sf E}[/tex]
To uncover the mystery phrase, place the letter from each answer into the corresponding boxes:
[tex]\Large\boxed{7}\;\;\boxed{1}\;\;\boxed{8}\;\;\boxed{8}\;\;\boxed{5}\;\;\boxed{2}\qquad \boxed{5}\;\;\boxed{6}\;\;\boxed{3}\;\;\boxed{8}\;\;\boxed{4}\\\\ \sf \;\:C \;\;\;\;\:H\;\;\;\:E\;\;\;\:\:E\;\;\;\:\:S\;\;\;\;\;Y\qquad \;\;\;F\;\;\;\;\;R\;\;\;\;\;\:I\;\;\;\;\;E\;\;\;\;\;S[/tex]
