Exponents and Roots: Tips and hints

DEFINITION - EXPONENTSExponents 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 - EXPONENTS1. 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 - ROOTSRoots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

TIPS - ROOTSGeneral 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.

Please share your Exponents and Roots tips below and get kudos point. Thank you. _________________

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