Power of a Number

Here we will learn the Power of a Number.

We know a × a = a², a × a × a = a³, etc., and a × a × a × .... n times = aⁿ, where n is a positive integer.

aⁿ 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 aⁿ where n is a positive rational number. Thus, aⁿ is a power of a for all values of n where n = positive/negative integer or positive/negative fraction. In fact, aⁿ is a power of a where n is any real number.

Examples on Power of a Number:

1. Find the value of 4⁵.

Solution:

4⁵ = 4 × 4 × 4 × 4 × 4 = 1024.

2. Find the value of 4^-5.

Solution:

4^-5 = Reciprocal of 4⁵ = $\frac{1}{4^{5}}$ = $\frac{1}{4 × 4 × 4 × 4 × 4}$ = $\frac{1}{1024}$.

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 4⁵ = 4^5/5 = 4.

Note: If a > 0, a ≠ 1, aⁿ 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 aⁿ we assume a > 0, a ≠ 1 and n is any real number.

9th Grade Math

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