🔢 Quant

Exponents, Roots & Decimals

Exponent rules, common-base rewriting, roots, place value and scientific notation — calculator-free number handling.

~5h

to master

Why exponents are everywhere

With no calculator in Quant, the GMAT can't ask you to compute big powers — it asks you to restructure them. Nearly every exponent question is solved by forcing both sides onto a common base, or by pulling a common factor out of a sum.

The exponent rules

📐Core Rule

For any nonzero base: $a^m \cdot a^n = a^{m+n}$ · $\dfrac{a^m}{a^n} = a^{m-n}$ · $(a^m)^n = a^{mn}$ · $a^0 = 1$ · $a^{-n} = \dfrac{1}{a^n}$ · $a^{1/2} = \sqrt{a}$ .

Two behaviours worth feeling in your bones:

Squaring grows numbers above 1 and shrinks numbers between 0 and 1: $3^2 = 9 > 3$ , but $(0.1)^2 = 0.01 < 0.1$ . Many trap options exploit exactly this.
Adding powers ≠ multiplying powers. $2^{10} + 2^{10} = 2 \cdot 2^{10} = 2^{11}$ , not $2^{20}$ . Sums of powers are handled by factoring: $3^{12} + 3^{10} = 3^{10}(3^2 + 1) = 10 \cdot 3^{10}$ .

✏️Worked Example

Simplify $\dfrac{6^8}{12^4}$ . Prime-factorize the bases:

6^8 = 2^8 3^8

and

12^4 = (2^2 \cdot 3)^4 = 2^8 3^4

. The quotient is

3^4 = 81

. Whenever bases differ, primes are the common language. See it in action: .

Roots

Every positive number has two square roots ( $\pm$ ), but the symbol $\sqrt{x}$ means the nonnegative one; $\sqrt{x^2} = |x|$ .
Cube roots keep the sign: $\sqrt[3]{-8} = -2$ . Square roots of negatives are not real — and the GMAT stays inside the reals.
Simplify by extracting square factors: $\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}$ .

Decimals, place value, scientific notation

In scientific notation, exactly one nonzero digit sits left of the point: $0.0231 = 2.31 \times 10^{-2}$ . The exponent counts how far the decimal point travels (right if positive, left if negative).
Multiplying decimals: multiply as integers, then place the point so the answer has as many decimal digits as the inputs combined ( $2.09 \times 1.3$ : $209 \times 13 = 2717$ , three decimal digits → $2.717$ ).
Dividing by a decimal: slide both points right until the divisor is an integer ( $698.12 / 12.4 = 6981.2 / 124$ ).
A repeating decimal is shown with a bar over the repeating block; the block repeats forever with nothing after it.

⚡Shortcut

Powers of 10 do the heavy lifting.

\dfrac{0.0048 \times 10^5}{0.12} = \dfrac{4.8 \times 10^2}{1.2 \times 10^{-1}} = 4 \times 10^{3}

. Convert everything to

a \times 10^k

form first; the arithmetic collapses to single digits. Try for the cycling version of this idea.

⚠️GMAT Trap

$(-3)^2$ and $-3^2$ are different numbers. The first is $9$ ; the second is $-(3^2) = -9$ . Exponents bind before the minus sign unless parentheses say otherwise. On the GMAT this single convention decides whole questions.

Checklist

Different bases? Rewrite in primes
Sum of powers? Factor out the smallest power
$\sqrt{x^2} = |x|$ , not $x$
Decimal point bookkeeping: count digits, don't guess
Scientific notation before multiplying/dividing ugly decimals

Sample Questions

22 practice questions

Medium

If $27^{x} = 9^{\,x+4}$ , what is the value of $x$ ?

Hard

If the integer $N = 3^{43} \times 7^{22}$ , what is the units digit of $N$ ?

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