Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{To rationalise }\frac{5}{2\sqrt{3}}\text{, you must multiply by }2\sqrt{3}\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False (mostly inefficient)} \\ \text{✅ Better Method: Multiply by }\sqrt{3}\text{ only.} \end{gathered}$$
• $\text{To simplify }\sqrt{72}\text{, which factors of 72 should you choose?}$
  $36\times 2$
• $$\begin{gathered} \text{True or False?} \\ \sqrt{a+b}=\sqrt{a}+\sqrt{b} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: It cannot be simplified separately.} \end{gathered}$$
• $\text{When simplifying a surd like }\sqrt{72}\text{, which pair of factors should you choose?}$
  $\text{The pair containing the largest square number.}$
• $\text{What do you multiply the numerator and denominator by to rationalise }\frac{1}{3+\sqrt{2}}\text{?}$
  $3-\sqrt{2}$
• $\text{How do you rationalise a denominator in the form }a+\sqrt{b}\text{?}$
  $\text{Multiply the numerator and denominator by the conjugate }a-\sqrt{b}\text{.}$
• $$\begin{gathered} \text{True or False?} \\ \text{If }f(x)=\sqrt{3x}\text{, then the index form is:} \\ f(x)=3x^{\frac{1}{2}}\text{?} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer:} \\ f(x)=\sqrt{3}{x}^{\frac{1}{2}}\text{ or }{3}^{\frac{1}{2}}{x}^{\frac{1}{2}} \\ \text{The square root applies to the }3\text{ as well. If differentiating, the coefficient is }\sqrt{3}\text{, not }3\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ (-5{)}^{-2}=25 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\frac{1}{(-5{)}^2}=\frac{1}{25} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ 2x^{-3}=\frac{1}{2x^3} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\frac{2}{x^3} \end{gathered}$$
• $\text{Rewrite }{x}^{\frac{m}{n}}\text{ using root notation.}$
  $\sqrt[n]{x^m}\text{ or }(\sqrt[n]{x}{)}^m$
• $$\begin{gathered} \text{True or False?} \\ (3x^2{)}^3=3x^6 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }27x^6 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \frac{8x^5}{2x^{-2}}=4x^3 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }4x^7 \end{gathered}$$
• $\text{Condition required to use the rule }{a}^m\times a^n=a^{m+n}\text{.}$
  $\text{The base (}a\text{) must be the same.}$
• $\text{Rewrite }{a}^{\frac{m}{n}}\text{ using root notation.}$
  $\sqrt[n]{a^m}\text{ or }(\sqrt[n]{a}{)}^m$
• $\text{First step to find the points of intersection of }y=x^2\text{ and }y=x+2\text{?}$
  $\text{Equate the two expressions: }{x}^2=x+2\text{.}$
• $$\begin{gathered} \text{True or False?} \\ \text{The solution to }{x}^2=9x\text{ is just }x=9\text{.} \end{gathered}$$
  $$\begin{gathered} \text{False} \\ \text{Correct Answer: }x=0\text{ and }x=9 \end{gathered}$$
• $\text{First step to solve }{x}^4-3x^2-4=0\text{?}$
  $\text{Use a substitution, e.g., let }u=x^2\text{.}$
• $\text{When solving }{x}^2=5x\text{, why is dividing both sides by }x\text{ risky?}$
  $\text{You lose the solution }x=0\text{.}$
• $$\begin{gathered} \text{True or False?} \\ \text{To solve }(x-3{)}^2=25\text{, the next step is }x-3=5\text{.} \end{gathered}$$
  $$\begin{gathered} \text{False} \\ \text{Correct Answer: }x-3=\pm 5 \end{gathered}$$
• $\text{What is the substitution strategy for solving }{x}^4-5x^2+4=0\text{?}$
  $\text{Let }u=x^2\text{, solve the quadratic for }u\text{, then find }x\text{.}$
• $$\begin{gathered} \text{What is the **first step** to integrate the following expression?} \\ \int \frac{(x+2{)}^2}{\sqrt{x}} \\ ,dx \end{gathered}$$
  $$\begin{gathered} \text{**Simplify the integrand** into a sum of terms in the form }a{x}^n\text{.} \\ \text{1. **Expand** the numerator: }{x}^2+4x+4 \\ \text{2. **Split** the fraction: }\frac{x^2}{x^{0.5}}+\frac{4x}{x^{0.5}}+\frac{4}{x^{0.5}} \\ \text{3. **Simplify** indices: }{x}^{1.5}+4x^{0.5}+4x^{-0.5} \end{gathered}$$
• $$\begin{gathered} \text{How do you write }\int \frac{1}{2\sqrt{x}} \\ ,dx\text{ in a form suitable for integration?} \end{gathered}$$
  $$\begin{gathered} \text{Convert the root to a fractional power and move it to the numerator:} \\ y=\frac{1}{2}{x}^{-\frac{1}{2}} \\ \text{Then apply the power rule.} \end{gathered}$$