Wednesday
Jun212017

Official ACT Prep Guide 2018: Almost identical to last year

The new Official ACT Prep Guide, 2018 1st Edition has been released in May 2017. This guide is almost identical to last year's version. All three of the practice tests are identical. The only change has been the replacement of Question 59 in Practice Test 1. The question on the left in the images shown below has been replaced by the question on the right in the 2018 ACT Guide. There was an error in the original question which I discussed in an earlier blog post: Error in Q59 of 2016 Official ACT Guide.

Apart from this minor change, the rest of my comments listed below are the same as I made last year in the review of 2016 Official ACT Guide.

  • The practice tests are a patchwork of questions from the the 5 tests in the Real ACT Guide and recently released ACT tests from 2015.
  • First Practice Test: The math section in the first practice test in the guide has around 40 questions from the June 2015 ACT test (Form 73C), and the remainder are a combination of new questions and questions from the five tests in the Real ACT Guide.
  • Second Practice Test: Half of the math questions are from the Test#2 from Real ACT guide and the remainder are either new or from the other tests in the Real ACT guide.
  • Third Practice Test: Half of the math questions are from April 2015 ACT test (Form 73G) and the rest are from Test#4 from the Real ACT Guide.
  • Because they have reused part of the tests that are in the older Real ACT guide, this makes the practice tests in the older guide useless. I don't know if this was done deliberately to ensure "planned obsolescence" of the older guide.

My final recommendation would be to use the April and June 2015 ACT tests on their own, and use the five practice tests in the Real ACT Guide, 3rd Edition for additional practice. I do not recommend buying this guide.

As I have said before, this analysis is based on the math section only, but I am pretty certain it will be mirrored in the remaining sections of the tests as well. I will leave it to others to analyze those sections.

Sunday
May212017

2017 April ACT Form 74F: Video Explanations

I have posted video explanations to all of the math questions in the 2017 April ACT Form 74F Test.

Saturday
Apr292017

ACT Math Practice Question: Simplifying algebraic fractions

Try this ACT math practice question on simplifying and rewriting algebraic fractions.:

If $a$, $b$, and $a-\dfrac{1}{b}$ are not 0, then $$ \dfrac{\dfrac{1}{a}-b}{a-\dfrac{1}{b}} $$ equals
  1. \(\quad -\dfrac{a}{b}\)

  2. \(\quad -\dfrac{b}{a}\)

  3. \(\quad 1\)

  4. \(\quad \dfrac{a}{b}\)

  5. \(\quad \dfrac{b}{a}\)



Sunday
Mar122017

Exponents: All the rules you need to know for ACT Math

Exponents: All you need to know for ACT Math

Here is a brief summary of all the exponent rules and manipulations that you are expected to know for the math section of the ACT test. You need to memorize these rules, and ensure that you can apply them in different situations.

Exponent Rules

$$ \begin{array}{|c|c|c|} \hline \textbf{Rule} & \textbf{Arithmetic example} & \textbf{Algebraic example} \\ \hline (a^m) (a^n) = a^{m+n} & (5^{3})(5^5) = 5^{3+5}=5^8 & (x^6)(x^{-4})=x^{6+(-4)} = x^2 \\ \hline (a^m)^n = a^{mn} = (a^n)^m &(2^2)^3=2^{6}=64 &(3z^2)^3=(3^3)(z^2)^3=27z^{6} \\ \hline a^{-n} = \dfrac{1}{a^n} \quad (a \neq 0)& 2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8} & x^{-3} = \dfrac{1}{x^3} \\ \hline \dfrac{a^m}{a^n} = a^{m-n} & \dfrac{7^{8}}{7^5} = 7^{8-5}=7^3 & \dfrac{x^5}{x^{-4}}=x^{5-(-4)} = x^9\\ \hline a^0=1 & (-5)^0=1 & x^0=1 \quad (x \neq 0) \\ \hline (a \times b)^n = (a^n)(b^n) & (2\times 5)^6=(2^6)(5^6) =10^6 & (2x)^3 = (2)^3(x)^3= 8x^3\\ \hline \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} & \left(\dfrac{3}{2}\right)^3 = \dfrac{3^3}{2^3}=\dfrac{27}{8} & \left(\dfrac{2x}{3y^{2}}\right)^3 = \dfrac{2^3 x^3}{3^{3} (y^{2})^3}= \dfrac{8x^3}{27 y^{6}}\\ \hline \end{array} $$

Exponents: Algebraic Manipulation

You will be expected to rewrite and manipulate algebraic expressions containing exponent terms. Here I list some examples of common manipulations: $$8^x = (2^3)^x = 2^{3x}$$ $$27^x = (3^3)^x = 3^{3x}$$ $$3^{6x} = (3^2)^{3x} = 9^{3x}$$ $$3^{6x} = (3^3)^{2x} = 27^{2x}$$ $$9(3^x) = (3^2)(3^x) = 3^{x+2} \quad \text{Note:} \quad 9(3^x) \neq 27^{x}$$ $$8(2^x) = (2^3)(2^x) = 2^{x+3} \quad \text{Note:} \quad 8(2^x) \neq 16^{x}$$ $$\dfrac{3^{x+1}}{3} =\dfrac{3^{x+1}}{3^1}= 3^{(x+1)-1} = 3^x \quad \text{Note:} \quad \dfrac{3^{x+1}}{3} \neq x+1$$

Exponents: Adding and Subtracting Terms

You will be asked to simplify exponent expressions such as $2^{8}+2^{8}$. The most common mistake students make is to add the exponents, $2^8 + 2^8 \neq 2^{16}$, instead $2^8 + 2^8 = 2^{9}$. There is no general exponent rule when adding powers of numbers that have the same base, however, there are cases where simplification is possible using other rules of arithmetic. If you see a question on the ACT that asks you to add exponent terms with the same bases, the best approach is to factor the largest common term which will almost always lead to simplification. For example: $$2^8 + 2^8 = 2^8(1 + 1) = 2^8(2) = 2^8(2^1) = 2^{8+1} = 2^9$$

Additional Examples

$$ 2^4 + 2^4 + 2^4 + 2^4 = 2^4(1 + 1 + 1 + 1) = 2^4(4) = 2^4(2^2) = 2^{4+2} = 2^6 $$ $$ 2^{20} - 3(2^{18}) = 2^{18}(2^2 - 3) = 2^{18}(4-3) =2^{18} $$ $$ 2^{x} + 2^{x+1} = 2^x + (2^{x})(2^1) = 2^{x}(1+2) =3(2^x)$$ $$\dfrac{1}{2^{9}} - \dfrac{1}{2^{10}} = \dfrac{1}{2^{9}}\left(1 - \dfrac{1}{2}\right) = \dfrac{1}{2^{9}}\left(\dfrac{1}{2}\right) = \dfrac{1}{2^{10}} \quad \text{Or} \quad \dfrac{1}{2^{9}} - \dfrac{1}{2^{10}} = \dfrac{2}{2^{10}} - \dfrac{1}{2^{10}} = \dfrac{2-1}{2^{10}} = \dfrac{1}{2^{10}}$$

Radicals and Fractional Exponents

  • A square root of a number $n$ is a number that, when squared, is equal to $n$ or $(\sqrt{n})^2 = n$.
  • An exponent of $\dfrac{1}{2}$ is the same as taking the square root of a number, $\sqrt{n} = n^{\frac{1}{2}}$.
  • A cube root of a number $n$ is a number that, when cubed, is equal to $n$ or $(\sqrt[3]{n})^3 = n$.
  • An exponent of $\dfrac{1}{3}$ is the same as taking the cube root of a number, $\sqrt[3]{n} = n^{\frac{1}{3}}$.
$$ \begin{array}{|c|c|} \hline \textbf{Rule} & \textbf{Example} \\ \hline (\sqrt{a})^2 = a \quad (a>0) & (\sqrt{7})^2 = 7 \\ \hline \sqrt{a^2} = a \quad (a>0) & \sqrt{7^2} = 7 \\ \hline \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} & \sqrt{6}\sqrt{8}=\sqrt{48} = \sqrt{(16)(3)} = \sqrt{16}\sqrt{3} = 4\sqrt{3} \\ \hline \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} & \dfrac{\sqrt{6}}{\sqrt{24}} = \sqrt{\dfrac{6}{24}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2} \\ \hline a^{\frac{m}{n}}= (a^{\frac{1}{n}})^m = (a^m)^{\frac{1}{n}}= \sqrt[n]{a^m} & 27^{\frac{2}{3}} = (27^2)^{\frac{1}{3}}=(27^{\frac{1}{3}})^2 = (\sqrt[3]{27})^2= (3)^2=9\\ \hline \end{array} $$ Here is a list of common exponent manipulations that you can expect on the ACT math section: $$ \sqrt[6]{27} = 27^{\frac{1}{6}} = (3^3)^{\frac{1}{6}} = 3^{\frac{3}{6}} = 3^{\frac{1}{2}} = \sqrt{3}$$ $$a^{\frac{3}{4}} = (a^{\frac{1}{4}})^3 = (\sqrt[4]{a})^3 = (a^3)^{\frac{1}{4}} = \sqrt[4]{a^3} $$ $$(\sqrt{x^{3}})^4 = [(x^{3})^{\frac{1}{2}}]^4 = x^{(3)(\frac{1}{2})(4)} = x^6$$ $$x \sqrt{x}=(x^1)(x^{\frac{1}{2}}) = x^{1+\frac{1}{2}} = x^{\frac{3}{2}}$$ $$ \sqrt[3]{8x^6} = (8x^6)^{\frac{1}{3}} = (8)^{\frac{1}{3}} (x^6)^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} (x)^{\frac{6}{3}} = 2x^2 $$ $$ \sqrt[6]{x^4 y^3 z^2} =(x^4 y^3 z^2)^{\frac{1}{6}} = x^{\frac{4}{6}} y^{\frac{3}{6}} z^{\frac{2}{6}} = x^{\frac{2}{3}} y^{\frac{1}{2}} z^{\frac{1}{3}} $$

Common Mistakes

Finally, I list the most common mistakes that students make when they have to simplify algebraic expressions with exponents. $$ \begin{array}{|c|c|} \hline \textbf{Mistake} & \textbf{Example} \\ \hline (3a)^3 \neq 3a^3 & \text{Instead} \quad (3a)^3 = (3^3)(a^3) = 27a^3 \\ \hline 3x^{-1} \neq \dfrac{1}{3x} & \text{Instead} \quad 3x^{-1} = \dfrac{3}{x} \\ \hline \sqrt{a^2 + b^2} \neq a + b & \sqrt{3^2 + 4^2} \neq 3 + 4 \quad \text{instead} \quad \sqrt{3^2+4^2}=\sqrt{25}=5 \\ \hline (a + b)^n \neq a^n + b^n & (2 + 3)^2 = 5^2 = 25 \neq 2^2 + 3^2 = 13 \\ \hline (a^m)(b^n) \neq (ab)^{m+n} & (2^3)(3^2) = (8)(9) = 72 \neq (2 \cdot 3)^{3+2} = (6)^5 = 7776\\ \hline (-a)^2 \neq -(a^2) & (-3)^2 = 9 \neq -(3^2) = -9 \quad \text{instead} \quad (-a)^2 = a^2 \\ \hline \end{array} $$
Thursday
Jan262017

2016 December ACT Form 74H: Video Explanations

I have posted video explanations to all of the math questions in the 2016 December ACT Form 74H Test.