Questions on exponents are considered to be freebies since they are relatively way easier than its more intelligent siblings like Permutations and Combinations, Geometry, Quadratics etc. However, that's the first to screwing up your GRE exam is to take anything lightly. Remember at the end of the day GRE is a fire-breathing dragon you have to slay and you cannot afford to have any blind-spots. In this post, we will try to provide a conceptually exhaustive rendition of exponents.

Basics: In an algebraic expression, \(y^a\), where x is raised to the power of a means

y * y * y * .... a times. here,

y is called the

base and

a is called the

exponent.

Some examples can be: \(2^4\)= 2*2*2*2 = 16 or \(5^x\) = 5 *5 *5 *... x times

The bases in the two examples are 2 and 5 respectively. The exponents are ... quite easy, you can decipher them yourselves.

Some rules of engagement with exponents are given below:

Laws of Exponents:- \(x^A * x^B = x^{(A+B)}\)

- \(\frac{x^A}{x^B} = x^{(A-B)}\)

- \(x^a * y^a = (xy)^a\)

- \(x^{(-a)} = \frac{1}{x^a}\)

- \(x^0 = 1\)

- \(x^1 = x\)

- \((x^A)^B = x^{(AB)}\)

- \(x^{\frac{a}{b}}\) = \(^b\sqrt{x^a}\)

The above rules are the exhaustive lists of concepts you can use to tackle any exponents problem. However, there are some tricky versions of the above rules which may fool you during your exams which are listed below:

Negative Bases

- \(x^A = +ve\) value; if \(x\) is -ve and \(A\) is even
- \(x^A = -ve\) value; if \(x\) is -ve and \(A\) is odd

For example: \((-3)^2 = 9\), \((-3)^3 = -27\)

Keep in mind \((-3)^2 = 9\), whereas \(-3^2 = -9\).

\(-3^2\) simply means negative of square of 3.

Square Roots

Square root is basically the inverse of a square. In terms of functions if we write,

\(f(x) = x^2\)

Then the inverse of the function will be

\(f^{-1}(x) = \sqrt{x}\)

Algebraic way of writing a square root is simply powering it to the power of \(\frac{1}{2}\), i.e,

\(x^{1/2} = \sqrt{x}\)

However, there are some pitfalls when it comes to square roots, like the following example:

- \(\sqrt{x} = r\); such that \(r^2 = x\), where \(r\) is nonnegetive.

Keep in mind eventhough \((-r)^2 = x\), symbol \(\sqrt{x}\) is used denote the nonegetive root of a number.

For example: \(\sqrt{9} = 3\); not -3.

Some basic properties of square roots are given below:

- \((\sqrt{x})^2 = x\)
- \(\sqrt{x^2} = x\)
- \(\sqrt{x}\sqrt{y} = \sqrt{xy}\)
- \(\sqrt{x}/\sqrt{y}= \sqrt{\frac{x}{y}}\)

_________________

Sandy

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