$\begin{array}{lll}& & \text{Let's Simplify Fractions}\\ & & \left[\right]\\ \text{•}& & \text{Find fractions to get lowest common denominator of}6\\ & & \left[\right]\\ \text{•}& & \text{Multiply}\\ & & \left[\right]\\ \text{•}& & \text{Add numerators}\\ & & \left[\right]\\ \text{•}& & \text{Multiply}3\cdot 1\\ & & \left[\right]\\ \text{•}& & \text{Multiply}1\cdot 1\\ & & \left[\right]\\ \text{•}& & \text{Add}3+1\\ & & \left[\right]\\ \text{•}& & \text{Cancel out factor of}2\\ & & \left[\right]\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Find fractions to get lowest common denominator of}6\\ & & \left[\right]\\ \text{•}& & \text{Multiply}\\ & & \left[\right]\\ \text{•}& & \text{Add fractions}\\ & & \frac{2}{3}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Pull out a factor of}\sqrt{4}=2\text{from}\sqrt{12}\\ & & \left[\right]\\ \text{•}& & \text{Multiply in order to rationalize the denominator}\\ & & \left[\right]\\ \text{•}& & \text{Multiply the denominator}\\ & & \left[\right]\\ \text{•}& & \text{Factor roots}\\ & & \left[\right]\\ \text{•}& & \text{Combine}\left\{\sqrt{2}\cdot \sqrt{2}\,\sqrt{3}\cdot \sqrt{3}\right\}\\ & & \left[\right]\\ \text{•}& & \text{Multiply}3\cdot \left(2\sqrt{5}\right)\\ & & \left[\right]\\ \text{•}& & \text{Cancel out factor of}6\\ & & \sqrt{5}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Multiply in order to rationalize the denominator}\\ & & \left[\right]\\ \text{•}& & \text{Multiply the denominator}\\ & & \left[\right]\\ \text{•}& & \text{Factor roots}\\ & & \left[\right]\\ \text{•}& & \text{Combine}\left\{3^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot \sqrt{3}\,2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot \sqrt{2}\cdot \sqrt{2}\right\}\\ & & \left[\right]\\ \text{•}& & \text{Multiply}33^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\cdot \left(22^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{5}\right)\\ & & \left[\right]\\ \text{•}& & \text{Multiply}2\cdot \left(6\sqrt{5}{2}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)\\ & & \left[\right]\\ \text{•}& & \text{Cancel out factor of}12\\ & & \sqrt{5}{2}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Evaluate exponent}{\left(3x{y}^2\right)}^2\\ & & \left[\right]\\ \text{•}& & \text{Multiply}2\cdot x^3\cdot y^3\cdot \left(9x^2{y}^4\right)\\ & & \left(18x^5{y}^7\right)\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply the product rule}{a}^n{a}^m=a^{n+m}\text{to add exponents with common base}\\ & & \left[\right]\\ \text{•}& & \text{Add exponents}\\ & & \left[\right]\\ \text{•}& & \text{Solution}\\ & & \left[\right]\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply the integer power of a power rule,}{\left(a^n\right)}^m=a^{nm}\\ & & \left[\right]\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Cancel out factor of}{y}^4\text{provided}{y}^4\ne 0\\ & & y\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Divide assuming}{y}^4\text{≠ 0}\\ & & \frac{1}{y^9}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply the quotient rule:}\frac{a^n}{a^m}=a^{n-m}\\ & & \left[\right]\\ \text{•}& & \text{Evaluate exponent}{123}^1\\ & & 123\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Use the log rule,}{\log }_a\left(x\right)=\frac{{\log }_b\left(x\right)}{{\log }_b\left(a\right)}\text{to express as a single logarithm}\\ & & \left[\right]\\ \text{•}& & \text{Solution}\\ & & \left[\right]\end{array}$

$\frac{\ln \left(2\right)}{\ln \left(5\right)}+\frac{\ln \left(3\right)}{\ln \left(2\right)+\ln \left(5\right)}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Evaluate}{\log }_{10}\left(100\right)\\ & & \left[\right]\\ \text{•}& & \text{Evaluate}{\log }_{10}\left(1000\right)\\ & & \left[\right]\\ \text{•}& & \text{Add}2+3\\ & & 5\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply the log rule}{\log }_a\left(m^n\right)=n{\log }_a\left(m\right)\\ & & \left[\right]\\ \text{•}& & \text{Apply the log rule}{\log }_a\left(a\right)=1\\ & & \left[\right]\\ \text{•}& & \text{Multiply}5\cdot 4\\ & & 20\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply}\text{Pythagoras}\text{trig identity,}{\cot \left(x\right)}^2={\csc \left(x\right)}^2-1\\ & & 1+\left({\csc \left(x\right)}^2-1\right)\\ \text{•}& & \text{Apply}\text{Reciprocal Function}\text{trig identity,}\csc \left(x\right)=\frac{1}{\sin \left(x\right)}\\ & & {\frac{1}{\sin \left(x\right)}}^2\\ \text{•}& & \text{Evaluate}\\ & & \frac{1}{{\sin \left(x\right)}^2}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Add}\mathrm{\pi }+\mathrm{\pi }+x\\ & & \left[\right]\\ \text{•}& & \text{Evaluate}\sin \left(2\mathrm{\pi }+x\right)\\ & & \sin \left(x\right)\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply}\text{Reciprocal Function}\text{trig identity,}\sec \left(x\right)=\frac{1}{\cos \left(x\right)}\\ & & \frac{{\sec \left(x\right)}^2-1}{{\frac{1}{\cos \left(x\right)}}^2}\\ \text{•}& & \text{Apply}\text{Pythagoras}\text{trig identity,}{\sec \left(x\right)}^2=1+{\tan \left(x\right)}^2\\ & & \left(\left(1+{\tan \left(x\right)}^2\right)-1\right){\cos \left(x\right)}^2\\ \text{•}& & \text{Apply}\text{Quotient}\text{trig identity,}\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}\\ & & {\frac{\sin \left(x\right)}{\cos \left(x\right)}}^2{\cos \left(x\right)}^2\\ \text{•}& & \text{Evaluate}\\ & & {\sin \left(x\right)}^2\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Apply}\text{Pythagoras}\text{trig identity,}{\cot \left(x\right)}^2={\csc \left(x\right)}^2-1\\ & & \left(-{\cos \left(x\right)}^2+1\right)\left(1+\left({\csc \left(x\right)}^2-1\right)\right)\\ \text{•}& & \text{Apply}\text{Reciprocal Function}\text{trig identity,}\csc \left(x\right)=\frac{1}{\sin \left(x\right)}\\ & & \left(-{\cos \left(x\right)}^2+1\right){\frac{1}{\sin \left(x\right)}}^2\\ \text{•}& & \text{Apply}\text{Pythagoras}\text{trig identity,}{\cos \left(x\right)}^2=1-{\sin \left(x\right)}^2\\ & & \frac{-\left(1-{\sin \left(x\right)}^2\right)+1}{{\sin \left(x\right)}^2}\\ \text{•}& & \text{Evaluate}\\ & & 1\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \left[\right]\\ \text{•}& & \text{Integral}\text{to evaluate}\\ & & \int x\ⅆx\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{power}}\text{rule to the term}\int x\ⅆx\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule, for n}\text{≠}\text{-1}\\ & & \int x^{\left[\right]}\ⅆx=\left[\right]\\ & \text{◦}& \text{This means:}\\ & & \int x\ⅆx=\left[\right]\\ & \text{◦}& \text{So,}\\ & & \int x\ⅆx=\frac{x^2}{2}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{x^2}{2}\\ \text{•}& & \text{Sub evaluated}\text{integral}\text{back in expression}\\ & & \frac{3x^2}{2}\end{array}$