Definitions
Using an exponent is a way of expressing multiplication of a number by itself. The exponent says how many times the number, called the base, is used as a factor. An expression that has a base and exponent is called a power.
Examples
52 = 5 × 5 = 25
base = 5, exponent = 2
63 = 6 × 6 × 6 = 216
base = 6, exponent = 3
34 = 3 × 3 × 3 × 3 = 81
base = 3, exponent = 4
Two exponents have special names. Numbers with the exponent 2 are called squared. The expression 52 can be read as “five to the second power” or “five squared.”
Numbers with the exponent 3 are called cubed. The expression 63 can be read as “six to the third power” or “six cubed.”
The expression 34 is read as “three to the fourth power.”
The reciprocal for a number with a positive exponent, 1/an (n is a positive integer), is equivalent to the number with the negative exponent, a–n.
\\[1ex]\textit{a}^{\displaystyle{-\textit{n}}} = \dfrac{1}{\textit{a}^{\displaystyle{\textit{n}}}}
Rules of Exponents
This list outlines the rules of exponents and shows an example of each.
1) a0 = 1, a ≠ 0
50 = 1
A number to the zero power equals 1.
00 is undefined.
2) a1 = a
71 = 7
A number to the first power is equal to itself.
3) am × an = am + n
32 × 33 = 32 + 3 = 35
x4 × x2 = x4 + 2 = x6
Add the exponents when multiplying two terms with the same base.
4) \dfrac{\textit{a}^{\displaystyle{\textit{m}}}}{\textit{a}^{\displaystyle{\textit{n}}}} = am – n, a ≠ 0
\dfrac{7^{\displaystyle{5}}}{7^{\displaystyle{3}}} = 75 – 3 = 72
\dfrac{\textit{y}^{\displaystyle{6}}}{\textit{y}^{\displaystyle{4}}} = y6 – 4 = y2
Subtract exponents when dividing two terms that have the same base.
5) (am)n= amn
(23)2 = 23 × 2 = 26
(y4)3 = y4 × 3 = y12
Multiply the exponents to raise a power to a power.
6) (ab)n= anbn
(3 × 5)2 = 32 52
(x2y)3 = (x2)3y3 = x6y3
To find the power of a product, distribute the power to each factor.
7) (\dfrac{\textit{a}}{\textit{b}})n = \dfrac{\textit{a}^{\displaystyle{\textit{n}}}}{\textit{b}^{\displaystyle{\textit{n}}}}, b ≠ 0
(\dfrac{3}{4})3 = \dfrac{3^{\displaystyle{3}}}{4^{\displaystyle{3}}}
(\dfrac{\textit{x}^{\displaystyle{5}}}{\textit{y}})2 = \dfrac{{(\textit{x}^{\displaystyle{5}})}^{{\displaystyle{2}}}}{\textit{y}^{\displaystyle{2}}} = \dfrac{\textit{x}^{\displaystyle{10}}}{\textit{y}^{\displaystyle{2}}}
The power of a quotient distributes the power to the numerator and denominator.
8) \textit{a}^{\displaystyle{\textit{n}}} = \dfrac{1}{\textit{a}^{\displaystyle{-\textit{n}}}}
52 = \dfrac{1}{5^{-\displaystyle{2}}}
A number with a positive exponent can be written as the reciprocal for a negative exponent.
Frequent Mistakes
Remember to use Experiment as a technique if you’re not sure you remember a rule. For example, here are some common mistakes shown with numbers used to test the rules.
an + am ≠ an + m
24 + 23 is not equal to 27.
16 + 8 ≠ 128
an – am ≠ an – m
24 – 23 is not equal to 2.
16 – 8 ≠ 2
an + bn ≠ (a + b)n
32 + 52 is not equal to (3 + 5)2.
9 + 25 ≠ 64
an + an ≠ a2n
72 + 72 is not equal to 74.
49 + 49 ≠ 49 × 49
an + an + an ≠ a3n
23 + 23 + 23 is not equal to 29.
8 + 8 + 8 ≠ 512
an × am ≠ anm
22 × 23 is not equal to 26.
4 × 8 ≠ 64
Exponents and Negative Numbers
When applying exponents to negative numbers, make sure your answer has the correct sign.
A negative number taken to an even exponent is always a positive number.
(-2)4 = (-2) × (-2) × (-2) × (-2) = 16
(-5)-2 = 1/(-5)2 = 1/((-5) × (-5)) = 1/25
A negative number taken to an odd exponent is always a negative number.
(-4)3 = (-4) × (-4) × (-4) = -64
(-2)-3 = 1/(-2)3 = 1/( (-2) × (-2) × (-2))
= 1/(-8) = -1/8
But a leading negative sign is not affected by the exponent.
-52 = -(25) = -25
-(-4)3 = -((-4) × (-4) × (-4)) = -(-64) = 64
Fractional Exponents
When a base is a non-negative number, it can be raised to a fractional power. Fractional exponents are the link between roots and exponents. When the exponent is a fraction, the numerator says what power to raise the base to, and the denominator says what root to take. You can raise the base to the power or take the root in either order.
Fractional exponents will be covered more completely in the next section, Roots.
When the numerator of the fractional exponent is 1, take the nth root of the base.
a1/n = \sqrt[\displaystyle{\textit{n}}]{\textit{a}}
The exponent 1/2 means take the square root of the base.
a1/2 = \sqrt{\textit{a}}
91/2 = \sqrt{9} = 3
The exponent 1/3 means take the cube root of the base.
a1/3 = \sqrt[\displaystyle{3}]{\textit{a}}
1251/3 = \sqrt[\displaystyle{3}]{125} = \sqrt[\displaystyle{3}]{5 × 5 × 5} = 5
The exponent m/n means raise the base to the power m and take the nth root.
Note, that m can be any integer, while n is a positive integer only.
am/n = \sqrt[\displaystyle{\textit{n}}]{\textit{a}^{\displaystyle{\textit{m}}}}
82/3 = \sqrt[\displaystyle{3}]{8^{\displaystyle{2}}} = \sqrt[\displaystyle{3}]{(2^{\displaystyle{3}})^{\displaystyle{2}}} = \sqrt[\displaystyle{3}]{2^{\displaystyle{6}}}
= 22 = 4
Simplifying Exponential Expressions
Looking at the answers before starting to solve the problem is a good strategy for questions with exponents. You may match the answer format by just simplifying or estimating, rather than calculating to get a value.
Simplifying may be all that is needed. Answers will often have exponents rather than a numerical value.
An estimate needs to match the accuracy of the possible answer choices.
For any answer format, the first step is often to factor then simplify using the rules of exponents.
Remember to use Experiment as a technique if you’re not sure you remember a rule.
Example
242 × 153 =
Solution
Answers will often contain the exponents rather than multiplying to get a value. Find the prime factors before multiplying the terms.
242 × 153
= (2 × 2 × 2 × 3)2 × (3 × 5)3
= (23 × 3)2 × (3 × 5)3
= (23)2 × 32 × 33 × 53
= 26 × 35 × 53
Example
Evaluate: 82/45
Solution
Find the prime factors to get the same base, then use the rule
\dfrac{\textit{a}^{\displaystyle{\textit{m}}}}{\textit{a}^{\displaystyle{\textit{n}}}} = am – n
\dfrac{8^{\displaystyle{2}}}{4^{\displaystyle{5}}} = \dfrac{{(2^{\displaystyle{3}})}^{{\displaystyle{2}}}}{{(2^{\displaystyle{2}})}^{{\displaystyle{5}}}} = \dfrac{2^{\displaystyle{6}}}{2^{\displaystyle{10}}}
= 26 – 10 = 2-4
To compare to the answer choices, you can evaluate further, to get a fraction or a decimal.
2-4 = \dfrac{1}{2^{\displaystyle{4}}} = \dfrac{1}{16} = 0.0625
Example
Simplify: [(2xy3× 9y] / [(3xy)2]
Solution
Remember that on the GRE, fractions might be on one line. So rewrite this as “built up”, then simplify.
Do the multiplication, then use the rule \dfrac{\textit{a}^{\displaystyle{\textit{m}}}}{\textit{a}^{\displaystyle{\textit{n}}}} = am – n for each base. (This is often called cancellation.)
\dfrac{{2\textit{xy}}^{\displaystyle{3}} × 9\textit{y}}{{(3\textit{xy})}^{\displaystyle{2}}} = \dfrac{18\textit{x}{\textit{y}^{\displaystyle{4}}}}{9{\textit{x}^{\displaystyle{2}}{\textit{y}^{\displaystyle{2}}}}} = \dfrac{2{\textit{y}^{\displaystyle{2}}}}{\textit{x}}Example
Evaluate: \dfrac{5}{3^{-\displaystyle{2}}}
Solution
Apply the rule a–n = \dfrac{1}{\textit{a}^{\displaystyle{\textit{n}}}} to the denominator.
\dfrac{5}{3^{-\displaystyle{2}}} = \dfrac{5}{\dfrac{1}{3^{\displaystyle{2}}}} …Remember that division by a fraction is multiplication by its reciprocal.
\dfrac{5}{\dfrac{1}{3^{\displaystyle{2}}}} = 5 × 32 = 45
Example
Factor.
(a) 72 + 75
(b) 32 + 122
Solution
Factor these by “undoing” using the distributive property.
(a) 72 + 75 = 72(1 + 73)
(b) 32 + 122 = 32 + (3 × 4)2
= 32 + 32 42 = 32(1 + 42)
Example
If x = 52 + 53 + 54 + 55, what is the greatest prime factor of x?
Solution
Factor to find all the prime factors.
52 + 53 + 54 + 55
= 52(1 + 5 + 52 + 53)
= 52 (1 + 5 + 25 + 125)
= 52 (156)
= 52 (2 × 2 × 3 × 13)
So the greatest prime factor is 13.
Example
A microorganism has a length of 5-7 meters. If 625 microorganisms of this length are arranged in a line, what is the length of the line in millimeters? (1m = 100 cm, 1 cm = 10 mm)
Solution
Factor 625.
625 = 252 = (52)2 = 54
Multiply. Use the rule
am × an = am + n
(5-7)(54) = 5-3 = 1/53 = 13/53
= (1/5) × (1/5) × (1/5)
= 0.2 × 0.2 × 0.2 = 0.008
Convert from meters to millimeters. 1 m = 100 cm, 1 cm = 10 mm, therefore
1 m = 1,000 mm (or 1 mm = 10-3 m), so multiply both sides by 1,000.
0.008 m = 8 mm
Common Values
These powers commonly appear on the GRE, so they should be memorized.
Squares
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82= 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
Cubes
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
Higher Powers
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1,024
32 = 9
33 = 27
34 = 81
35 = 243
52 = 25
53 = 125
54 = 625
102 = 100
103 = 1,000
104 = 10,000
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Exponents
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Knowing these values and the rules of exponents means you can more quickly calculate other values.
For example, since 210 = 1,024, you know 212 = 1,024 × 4 = 4,096 and
2-10 = 1/1,024.
Before attempting these problems, be sure to review this section on Quantitative Comparison questions.
https://www.youtube.com/watch?v=pR6TBRSNdCs&list=PLD0D070C218D8F5A3&index=32
https://www.youtube.com/watch?v=E2Fe5MtP2j0&list=PLD0D070C218D8F5A3&index=34
https://www.youtube.com/watch?v=B-GfQYkiOpg&list=PLD0D070C218D8F5A3&index=35
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