$\begin{array}{lll}& & \text{Let's Simplify Fractions}\\ & & \frac{1}{2}+\frac{1}{6}\\ \text{•}& & \text{Find fractions to get lowest common denominator of}6\\ & & \frac{3}{3}\cdot \left(\frac{1}{2}\right)+\frac{1}{1}\cdot \left(\frac{1}{6}\right)\\ \text{•}& & \text{Multiply}\\ & & \frac{3\cdot 1}{6}+\frac{1\cdot 1}{6}\\ \text{•}& & \text{Add numerators}\\ & & \frac{\left(3\cdot 1+1\cdot 1\right)}{6}\\ \text{•}& & \text{Multiply}3\cdot 1\\ & & \frac{3+1\cdot 1}{6}\\ \text{•}& & \text{Multiply}1\cdot 1\\ & & \frac{3+1}{6}\\ \text{•}& & \text{Add}3+1\\ & & \frac{4}{6}\\ \text{•}& & \text{Cancel out factor of}2\\ & & \frac{2}{3}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{1}{2}+\frac{1}{6}\\ \text{•}& & \text{Find fractions to get lowest common denominator of}6\\ & & \frac{3}{3}\cdot \left(\frac{1}{2}\right)+\frac{1}{1}\cdot \left(\frac{1}{6}\right)\\ \text{•}& & \text{Multiply}\\ & & \frac{3\cdot 1}{6}+\frac{1\cdot 1}{6}\\ \text{•}& & \text{Add fractions}\\ & & \frac{2}{3}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{\sqrt{6}\cdot \sqrt{10}}{\sqrt{12}}\\ \text{•}& & \text{Pull out a factor of}\sqrt{4}=2\text{from}\sqrt{12}\\ & & \frac{\sqrt{6}\sqrt{10}}{\left(2\sqrt{3}\right)}\\ \text{•}& & \text{Multiply in order to rationalize the denominator}\\ & & \frac{\sqrt{3}}{\sqrt{3}}\cdot \left(\frac{\sqrt{6}\sqrt{10}}{2\sqrt{3}}\right)\\ \text{•}& & \text{Multiply the denominator}\\ & & \frac{\sqrt{3}\cdot \left(\sqrt{6}\sqrt{10}\right)}{6}\\ \text{•}& & \text{Factor roots}\\ & & \frac{\sqrt{3}\cdot \left(\sqrt{2}\cdot \sqrt{3}\cdot \left(\sqrt{2}\cdot \sqrt{5}\right)\right)}{6}\\ \text{•}& & \text{Combine}\left\{\sqrt{2}\cdot \sqrt{2}\,\sqrt{3}\cdot \sqrt{3}\right\}\\ & & \frac{3\cdot \left(2\cdot \sqrt{5}\right)}{6}\\ \text{•}& & \text{Multiply}3\cdot \left(2\sqrt{5}\right)\\ & & \frac{\left(6\sqrt{5}\right)}{6}\\ \text{•}& & \text{Cancel out factor of}6\\ & & \sqrt{5}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{\sqrt{6}\cdot \sqrt{10}}{{12}^{\frac{1}{3}}}\\ \text{•}& & \text{Multiply in order to rationalize the denominator}\\ & & \frac{{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\cdot \left(\frac{\sqrt{6}\sqrt{10}}{{12}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)\\ \text{•}& & \text{Multiply the denominator}\\ & & \frac{{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot \left(\sqrt{6}\sqrt{10}\right)}{12}\\ \text{•}& & \text{Factor roots}\\ & & \frac{3^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot \left(22^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\cdot \left(\sqrt{2}\cdot \sqrt{3}\cdot \left(\sqrt{2}\cdot \sqrt{5}\right)\right)}{12}\\ \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\}\\ & & \frac{2\cdot \left(\left(33^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)\cdot \left(\left(22^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\cdot \sqrt{5}\right)\right)}{12}\\ \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)\\ & & \frac{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)}{12}\\ \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)\\ & & \frac{\left(12\sqrt{5}{2}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)}{12}\\ \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}$

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

$\begin{array}{lll}& & \text{Let's simplify}\\ & & x^3\cdot x^5\\ \text{•}& & \text{Apply the product rule}{a}^n{a}^m=a^{n+m}\text{to add exponents with common base}\\ & & x^{3+5}\\ \text{•}& & \text{Add exponents}\\ & & x^8\\ \text{•}& & \text{Solution}\\ & & x^8\end{array}$

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

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{y^5}{y^4}\\ \text{•}& & \text{Cancel out factor of}{y}^4\text{provided}{y}^4\ne 0\\ & & y\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{y^{\mathrm{-5}}}{y^4}\\ \text{•}& & \text{Divide assuming}{y}^4\text{≠ 0}\\ & & \frac{1}{y^9}\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{{123}^5}{{123}^4}\\ \text{•}& & \text{Apply the quotient rule:}\frac{a^n}{a^m}=a^{n-m}\\ & & {123}^1\\ \text{•}& & \text{Evaluate exponent}{123}^1\\ & & 123\end{array}$

$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{{\log }_{10}\left(2\right)}{{\log }_{10}\left(5\right)}+{\log }_{10}\left(3\right)\\ \text{•}& & \text{Use the log rule,}{\log }_a\left(x\right)=\frac{{\log }_b\left(x\right)}{{\log }_b\left(a\right)}\text{to express as a single logarithm}\\ & & \left({\log }_5\left(2\right)\right)+{\log }_{10}\left(3\right)\\ \text{•}& & \text{Solution}\\ & & {\log }_5\left(2\right)+{\log }_{10}\left(3\right)\end{array}$

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

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

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

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

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

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

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

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