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The pdf file above contains all of the questions published in the GMATQuantum blog. Please download the file and then open it in Adobe Acrobat Reader for all of its features to work properly.

Tuesday
Mar212017

## Exponents: All you need to know for GMAT test

In this post I first summarize all the exponent rules that you need to know for the GMAT, followed by how these exponent rules are tested on the GMAT.

Exponent Rules: Summary

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

Exponents: Arithmetic and Algebraic Manipulation

You will be expected to rewrite and manipulate arithmetic and algebraic expressions containing exponent terms. Here I list examples of common exponent manipulations tested on the GMAT: $$8^x = (2^3)^x = 2^{3x}$$ $$27^x = (3^3)^x = 3^{3x}$$ $$3^{6x} = (3^2)^{3x} = 9^{3x}$$ $$3^{6x} = (3^3)^{2x} = 27^{2x}$$ $$9(3^x) = (3^2)(3^x) = 3^{x+2} \quad \text{Note:} \quad 9(3^x) \neq 27^{x}$$ $$8(2^x) = (2^3)(2^x) = 2^{x+3} \quad \text{Note:} \quad 8(2^x) \neq 16^{x}$$ $$\dfrac{9^x}{27^y} = \dfrac{(3^2)^x}{(3^3)^y} = \dfrac{3^{2x}}{3^{3y}} = 3^{2x-3y}$$ $$\dfrac{3^{x+1}}{3} =\dfrac{3^{x+1}}{3^1}= 3^{(x+1)-1} = 3^x \quad \text{Note:} \quad \dfrac{3^{x+1}}{3} \neq x+1$$ $$2^{8} \times 5^{9} = 2^{8} \times 5^{8} \times 5^{1} = (2 \times 5)^8 \times 5 = 5(10^8)$$ $$3(6)^8 = 3(2 \times 3)^8 = 3 \times 2^8 \times 3^8 = 2^8 \times 3^8 \times 3^1 = 2^8 \times 3^9$$ $$(2^{32})(3^{31}) + (2^{31})(3^{32}) + (2^{31})(3^{31}) = 2^{31}3^{31} (2^{1} + 3^{1} + 1) = (2 \times 3)^{31}(6) = (6)^{31}(6)^{1} = 6^{32}$$

Exponents: Sum and Difference of Powers

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. In general, if you see a question on the GMAT that asks you to add terms with the same bases, the best approach is to factor the largest common term, and in most cases the resulting terms will collapse to something simple. $$2^{22} + 2^{22} = 2^{22}(1 + 1) = 2^{22}(2) = (2^{22})(2^{1}) = 2^{22+1} = 2^{23}$$ $$2^4 + 2^4 + 2^4 + 2^4 = 2^4(1 + 1 + 1 + 1) = 2^4(4) = 2^4(2^2) = 2^{4+2} = 2^6$$ $$3^{33} + 3^{33} + 3^{33} = 3^{33}(1 + 1+1) = 3^{33}(3) = (3^{33})(3^{1}) = 3^{33+1} = 3^{34}$$ $$7^9 - 7^8 = 7^8(7 - 1) = 6(7^8)$$ $$\require{cancel} \displaystyle \frac{10^{11}+10^{12}+10^{13}}{10^6+10^7+10^8} = \frac{10^{11}(1+10+10^{2})}{10^{6}(1+10+10^2)} = \frac{10^{11}\cancel{(1+10+10^{2})}}{10^{6}\cancel{(1+10+10^2)} } =\frac{10^{11}}{10^6}=10^{11-6}=10^{5}$$ $$2^{x} + 2^{x+1} = 2^x + (2^{x})(2^1) = 2^{x}(1+2) =3(2^x)$$ $$5^{x} - 5^{x-2} = 5^{x-2}(5^2 - 1) = 24(5^{x-2})$$ $$\dfrac{1}{2^{9}} - \dfrac{1}{2^{10}} = \dfrac{1}{2^{9}}\left(1 - \dfrac{1}{2}\right) = \dfrac{1}{2^{9}}\left(\dfrac{1}{2}\right) = \dfrac{1}{2^{10}}$$ $$\dfrac{1}{2^{8}} + \dfrac{1}{2^{9}} + \dfrac{1}{2^{9}} = \dfrac{1}{2^{9}}\left(2 + 1 + 1\right) = \dfrac{4}{2^{9}} = \dfrac{2^2}{2^{9}} = \dfrac{1}{2^7}$$ $$\dfrac{1}{2^{x}} + \dfrac{1}{2^{x}} = \dfrac{2}{2^{x}} = \dfrac{1}{2^{x-1}}$$

Exponents: Common Mistakes

• $$(2^4)(2^4) \neq 2^{16}$$, instead $$(2^4)(2^4)=2^{4+4} = 2^{8}$$.
• $$(2^{2})(2^{x}) \neq 2^{2x}$$, instead $$(2^{2}) (2^{x}) = 2^{2+x}$$.
• $$(3^{x})^2 \neq (3^{x^2})$$, instead $$(3^{x})^2 = 3^{2x}$$.
• $$10^8-10^7 \neq 10^1$$, instead $$10^8 - 10^7 = 10(10^7) - 10^7 = 10^7(10-1) = 9(10^7)$$.
• $$[3^{x-2}]^3 \neq 3^{(3x-2)}$$, instead $$[3^{x-2}]^3 = [3^{3(x-2)}]= 3^{3x-6}$$.
• $$\displaystyle \frac{2^{22} + 4^{11}}{2^{22}} \neq \frac{\cancel{2^{22}} + 4^{11}}{\cancel{2^{22}}} = 4^{11}$$, instead $$\displaystyle \frac{2^{22} + 4^{11}}{2^{22}} = \frac{{2^{22}} + (2^2)^{11}}{2^{22}} = \frac{{2^{22}} + 2^{2\times 11}}{2^{22}}= \frac{2^{22} + 2^{22}}{2^{22}}= \frac{2^{22}(1 + 1)}{2^{22}} = \frac{\cancel{2^{22}}(2)}{\cancel{2^{22}}} = 2$$
• $$20(20^{599}) \neq 400^{599}$$, instead $$20(20^{599})=20^{600} = (20^{2})^{300} = 400^{300}$$.
Thursday
Mar092017

## GMAT Problem Solving 83: Hexagon and familiar triangles

Try this GMAT problem solving on hexagons and how to create familiar right triangles to find the length of sides. The video explanation is attached after the problem statement.

Wednesday
Feb082017

## GMAT Data Sufficiency: Consistency of statements

In this video lesson I go over the consistency of the two statements in data sufficiency, and how that can help you avoid falling for trap answers. I discuss two problems in the video, and I have listed them below. Please attempt these two problems and then review my video lesson.

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