Week 4 · ABSTRACT THINKING · 20–25 hrs

Exponents, inequalities & CR reasoning

Exponents and roots

Quantitative

GMAT Focus seldom asks you to simplify a power. It hides exponent structure inside comparisons, number-property constraints and Data Sufficiency traps.

The common disguise is a question that never says "exponent": comparing $2^{30}$ with $3^{20}$, counting trailing zeros of a factorial, or finding a units digit. The move is mechanical — force a common base or a common exponent. For the comparison, rewrite as $(2^3{)}^{10}=8^{10}$ against $(3^2{)}^{10}=9^{10}$ and read it off.

Three decisions separate the credited answer from the tempting one:

• An even root is not the inverse of squaring. $x^2=81$ has two real solutions; $\sqrt{81}$ has one. The real test is whether the stem's constraint (integer, positive, non-zero) kills the negative branch.
• $\sqrt{x^2}=\mathrm{\mid }x\mathrm{\mid }$, not $x$. Substituting a negative number to check is faster than reasoning about it.
• Constraints travel with the variable. "$n$ is a positive integer" and "$n$ is a positive number" produce different answer counts on the same-looking DS stem.

The reliable loss here is mechanical: distributing a power over a sum, $(a+b{)}^2\mathrm{\ne }{a}^2+b^2$, and dropping the negative root whenever the stem hands you an even power.

Worth having: $a^m{a}^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$, valid for $a>0$; for $a<0$ they hold only when the exponents are integers.

Inequalities and the sign of the unknown

Quantitative

On this test, inequalities are not a manipulation exercise; they are a sign-management exercise. Nearly every hard question turns on the fact that the algebraic move you want to make is legal only when the unknown has a known sign — and the question is usually built so that it doesn't.

The load-bearing rule: for $a>b$, multiplying by $c$ preserves the inequality if $c>0$ and reverses it if $c<0$. Nothing else in the topic does as much damage.

Disguises: cross-multiplication in a ratio ($\frac{1}{x}>\frac{1}{y}$); comparing squares, where $x>y$ does not give $x^2>y^2$; "which of the following must be true" items where every option is an ordinary rearrangement; Data Sufficiency statements that pin down $\mathrm{\mid }x\mathrm{\mid }$ without pinning the sign of $x$.

Three decisions separate right from tempting:

• Never multiply or divide by anything containing the unknown unless the stem licenses its sign.
• Test $0$, a small positive, and a negative before committing — signs and boundaries, not magnitudes.
• In Data Sufficiency, a statement settles the sign only if it forces one: $x^2>4$ doesn't; $x>2$ does.

The mechanical loss: a candidate cross-multiplies in a DS item, reaches a clean expression, marks it sufficient, and the answer hinges on the negative case they never tested. The arithmetic was flawless; the sign was never checked.

Worth memorising: $\frac{1}{x}>1$ is equivalent to $0<x<1$, not to $x<1$.

Absolute value

Quantitative

GMAT Focus rarely asks you to compute an absolute value. It uses one as a sign-suppression device, and the actual question is whether you can tell when a quantity's sign is knowable and when it is not.

The workhorse reading is distance: $\mathrm{\mid }x-a\mathrm{\mid }$ is the distance from $x$ to $a$, so $\mathrm{\mid }x-3\mathrm{\mid }<2$ means "within 2 of 3". Reach for that before case-splitting, because it collapses a two-case problem into one interval. Worth having: $\mathrm{\mid }x\mathrm{\mid }<a⟺-a<x<a$, and $\mathrm{\mid }x\mathrm{\mid }>a⟺x<-a$ or $x>a$ — both only for $a>0$. Applying the first form with a negative right side is a silent error.

Disguises: $\sqrt{x^2}=\mathrm{\mid }x\mathrm{\mid }$; even powers such as $x^4=16$ or $x^2=(x-2{)}^2$; "how far apart are $p$ and $q$"; and any Data Sufficiency stem demanding the sign of a variable, where $\mathrm{\mid }x\mathrm{\mid }=x$ if and only if $x\ge 0$.

The decisions: is this a distance (solve geometrically, no cases) or a definition (two cases, take both)? Must the non-absolute side be non-negative — $\mathrm{\mid }2x-5\mathrm{\mid }=3x$ admits no negative root, so $x\ge 0$ frames everything? And does each candidate satisfy the case assumption it was solved under, not merely the equation you produced?

The mechanical point-loser is squaring both sides to kill the bars, then not substituting back. Squaring invents roots that satisfy the squared equation and not the original; test makers place those invented roots in the answer choices, usually as the tempting middle option.

Critical Reasoning: evaluate, flaw and paradox

Verbal

Evaluate, flaw and paradox questions rarely name themselves. They appear as ordinary arguments: a company's sales rose after a rebrand; a city's crime fell after a curfew; a study links a gene to a disease. What GMAT Focus tests is not your ability to spot the label but to locate the gap between evidence and conclusion—the unstated assumption, the causal leap, the scope shift.

The disguises are thin. A flaw question may ask why the argument is vulnerable to criticism; an evaluate question may ask what would most usefully be known; a paradox may present two facts as a surprising contrast. In each, you are hunting the same thing: what must be true for the conclusion to follow?

Three decisions separate correct from tempting. For evaluate, the answer must be a question whose answer could cut either way—strengthen or weaken—and would change the argument's force. For flaw, you must name the precise error: correlation treated as causation, a necessary condition treated as sufficient, a sample too small or unrepresentative to support the generalisation. For paradox, the resolution must reconcile both facts, usually by introducing a third factor that makes both true; explaining one away is wrong.

Strong candidates lose points by answering the question they expected. On evaluate they pick a strengthener or weakener; on paradox they discredit one fact; on flaw they accept a vague objection. The mechanical fix: before reading options, articulate the assumption and negate it. The argument's conclusion collapses if that negation is plausible, and that is your answer.

Two-Part Analysis

Data Insights

Two-Part Analysis is not two questions glued together. The exam gives a short scenario and two columns of options; you must choose one option per column so that the pair satisfies a joint condition. The condition is often an equation, but it can be logical: one column might be a cause, the other an effect; one a premise, the other a conclusion.

Disguises: the coupling hides in a clause you read past—'...the same amount...', '...twice as many...', '...if and only if...'. Or the two columns look like independent variables, and the options are arranged so that the best option in each column alone is a tempting pair that violates the link.

Three decisions separate the correct pair. First, before evaluating options, write the relation that binds the columns. Second, treat the columns as a system: if one choice is fixed, the other is usually forced, so use the more constrained column as your filter. Third, verify the pair against every constraint, not just the one you used to find it.

The mechanical loss: strong candidates solve one column correctly, then pick the other column's option that 'fits the story' rather than the one that satisfies the joint condition exactly. Or they assume independence and select the best option in each column separately.

Useful memory: if two quantities have sum $S$ and difference $D$, then $x=(S+D)\mathrm{/}2$ and $y=(S-D)\mathrm{/}2$ (with $x\ge y$). Use it when the question gives a total and a 'how many more' in any form.

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