Here we will learn the Power of a Number.

We know a × a = a^{2}, a × a × a = a^{3}, etc., and a × a × a × .... n times = a^{n}, where n is a positive integer.

a^{n} is a power of a whose base is a and the index of power is n.

a^{p/q} is the qth root of a^{p} if p, q are positive integers.

a^{-n} is the reciprocal of a^{n} where n is a positive rational number. Thus, a^{n} is a power of a for all values of n where n = positive/negative integer or positive/negative fraction. In fact, a^{n} is a power of a where n is any real number.

Examples on Power of a Number:

**1.** Find the value of 4^{5}.

**Solution:**

4^{5} = 4 × 4 × 4 × 4 × 4 = 1024.

**2.** Find the value of 4^{-5}.

**Solution:**

4^{-5} = Reciprocal of 4^{5} = \(\frac{1}{4^{5}}\) = \(\frac{1}{4 × 4 × 4 × 4 × 4}\) = \(\frac{1}{1024}\).

**3.** Find the value of (1024)^{1/5}.

**Solution:**

(1024)^{1/5} = 5^{th} root of 4 × 4 × 4 × 4 × 4 = 5^{th}
root of 4^{5} = 4^{5/5} = 4.

**Note:** If a > 0, a ≠ 1, a^{n} is an exponent of a (read as
exponential of a), where n is any real number.

Though positive or negative integral powers of positive or
negative numbers can be found, it becomes difficult or impossible to find the
value of any power of negative numbers in the set of real numbers. Hence, in
what follows, for a^{n} we assume a > 0, a ≠ 1 and n is any real number.

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