Exponents and Roots: Tips and hints
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DEFINITION - EXPONENTS
Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number a multiplied n times can be written as a^n, where a represents the base, the number that is multiplied by itself n times and n represents the exponent. The exponent indicates how many times to multiple the base, a, by itself.
TIPS - EXPONENTS
1. Exponents one and zero:
a^0=1 Any nonzero number to the power of 0 is 1.
For example: 5^0=1 and (-3)^0=1
Note: the case of 0^0 is not tested on the GMAT.
a^1=a Any number to the power 1 is itself.
2. Powers of zero:
If the exponent is positive, the power of zero is zero: 0^n = 0, where n > 0.
If the exponent is negative, the power of zero (0^n, where n < 0) is undefined, because division by zero is implied.
3. Powers of one:
1^n=1 The integer powers of one are one.
4. Negative powers:
a^{-n}=\frac{1}{a^n}
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, 0^{-1} = \frac{1}{0}=undefined.
5. Powers of minus one:
If n is an even integer, then (-1)^n=1.
If n is an odd integer, then (-1)^n =-1.
6. Operations involving the same exponents:
Keep the exponent, multiply or divide the bases
a^n*b^n=(ab)^n
\frac{a^n}{b^n}=(\frac{a}{b})^n
(a^m)^n=a^{mn}
a^m^n=a^{(m^n)} and not (a^m)^n (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)
7. Operations involving the same bases:
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
a^n*a^m=a^{n+m}
\frac{a^n}{a^m}=a^{n-m}
8. Fraction as power:
a^{\frac{1}{n}}=\sqrt[n]{a}
a^{\frac{m}{n}}=\sqrt[n]{a^m}
DEFINITION - ROOTS
Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.
TIPS - ROOTS
General rules:
1. \sqrt{x}\sqrt{y}=\sqrt{xy} and \frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}.
2. (\sqrt{x})^n=\sqrt{x^n}
3. x^{\frac{1}{n}}=\sqrt[n]{x}
4. x^{\frac{n}{m}}=\sqrt[m]{x^n}
5. {\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}
6. \sqrt{x^2}=|x|, when x\leq{0}, then \sqrt{x^2}=-x and when x\geq{0}, then \sqrt{x^2}=x
7. When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt[4]{x}, then the only accepted answer is the positive root.
That is, \sqrt{25}=5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.
8. Odd roots will have the same sign as the base of the root. For example, \sqrt[3]{125} =5 and \sqrt[3]{-64} =-4.
This week's PS question
This week's DS Question
Theory: math-number-theory-88376.html
All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60
All DS roots problems to practice: search.php?search_id=tag&tag_id=49
All PS roots problems to practice: search.php?search_id=tag&tag_id=113
Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html
This week's DS Question
Theory: math-number-theory-88376.html
All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60
All DS roots problems to practice: search.php?search_id=tag&tag_id=49
All PS roots problems to practice: search.php?search_id=tag&tag_id=113
Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html
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