Power of a Power Rule

The power of a power rule is among the most important exponent laws. It is mainly applied to simplify expressions in the form (x^a)^b. Mathematically, it can be represented as (x^a)^b = x^{a × b} = x^ab where the exponents are multiplied together.

The formula for the power of a power rule is (x^a)^b = x^ab where x is the base, and a and b are exponents. This formula is used to solve expressions like:

• (x³)² = x^{3 × 2} = x⁶
   
• (5⁵)³ = 5^{5 × 3} = 5¹⁵ 
   
• (x⁴)³ = x^{4 × 3} = x¹²

The same rule is applied even for expressions with negative exponents. In (x^a)^b, if a and b are less than 0, then both the exponents are negative. Therefore, the formulas will change accordingly: 

• (a^-m)^-n = a^{(-m) × (-n)} = a^mn
   
•  (a^-m)ⁿ = a^{(-m) × (n)} = a^-mn
   
• (a^m)^-n = a^{(m) × (-n)} = a^-mn

If the exponents are in the fractional form of p/q, where p and q are integers, then we can use the formula ((a^p/q)^m/n) to solve such expressions. Let us take a look at the formulas when the exponents are fractions:

• (x^m/n)^p/q = x^mp/nq
   
• (x^m)^p/q = x^pm/q
   
• (x^m/n)^p = x^pm/n

So far, we’ve learned about the power of a power rule. In this section, we will see how to simplify expressions using this rule. 

For example, simplify (5²)³.

The formula of the power of a power rule is: (x^a)^b = x^ab

Here, x = 5, a = 2, and b = 3

Substituting the values we get, (5²)³ = 5^{(2 × 3)}

= 5⁶

= 5 × 5 × 5 × 5 × 5 × 5 

= 15625

The objective of the power of a power rule is to simplify expressions with an exponent raised to another exponent. Here are some real-life applications: 

• Finance: As the formula for calculating compound interest is A = P(1 + (r/n))^nt, we use the power of the power rule in banks and financial institutions.   

• Physics: It is used to represent very large or very small numbers in astronomy, physics, or nanotechnology. For example, (5 × 10¹⁸)² can be written as 5 × 10³⁶. Scientific notations like 5 × 1036 are used in various scientific fields.

• Computing Power: In computer science, the rule is used to calculate nested exponential growth in computing.