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:
 

• \((x^3)^2 = x^{(3 × 2)} = x^6\)
   
• \((5^5)^3 = 5^{(5 × 3)} = 5^{15}\)
   
• \((x^4)^3 = x^{(4 × 3)} = x^{12}\)

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 \(\frac{p}{q}\), where p and q are integers, then we can use the formula \(((a^\frac {p}{q})^\frac {m}{n})\) to solve such expressions.

Let us take a look at the formulas when the exponents are fractions:

• \((x^{\frac{m}{n}})^{\frac{p}{q}} = x^{\frac {mp}{nq}}\)
   
• \((x^{m})^{\frac{p}{q}} = x^{\frac {pm}{n}}\)
   
• \((x^{\frac{m}{n}})^p = x^{\frac {pm}{n}}\)

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 \((5^2)^3\).

The formula of the power of a power rule is:

\((x^a)^b = x^{a × b} = x^{ab}\)

Here, \(x = 5\), \(a = 2\), and \(b = 3\)

Substituting the values we get,

\((5^2)^3 = 5^{(2 × 3)}\\ (5^2)^3= 5^6\\ (5^2)^3= 5 × 5 × 5 × 5 × 5 × 5 \\ (5^2)^3= 15625\)

Here are some of the basic tips and tricks for students to master in the power of a power rule.
 

1. Remember the keyword "multiply the powers." Do not perform any other operation like addition when using the power of a power rule.
    
2. Remember that only the power of a power multiplies exponents. Do not multiply the powers with numbers.
    
3. Try simple examples to build confidence. Once you’re comfortable, move to algebraic ones like \((x^2 y)^3\)
    
4. Apply the rule to real-life problems. Relate it with concepts like compound interest, population growth, scaling in 3D models, etc. Seeing it in context strengthens understanding. 
    

5. Practice mixed exponent rules. Combine rules to master exponent operations like:
   
   \(\frac {(a^2)^3}{a^4} = a^{6-4} = a^2\)
   
   This helps students in avoiding confusion when multiple rules appear together.

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: 
 

1. Data storage and file sizes: If you have 1 MB of data, and it grows by a factor of \(10^3\) (KB in an MB) and then again by \(10^3\) (MB in a GB):
   
   \((10^3)^2 = 10^{3×2} = 10^6\)
   
   That means \(1 GB = 10^6\) bytes, illustrating the power of a power rule in computing units.
    
2. Physics: It is used to represent very large or very small numbers in astronomy, physics, or nanotechnology. For example, \((5 × 10^{18})^2\) can be written as \(5 × 10^{36}\). Scientific notations like 5 × 1036 are used in various scientific fields.
    
3. Computing Power: In computer science, the rule is used to calculate nested exponential growth in computing. 
    
4. Compound interest in finance: If an investment’s value increases by a factor of 1.051.051.05 each month, and you consider 12 months per year for 5 years: 
   
         \((1.05^{12})^5 = 1.05^{60}\)
   
   This means the total growth over 5 years equals 60 months of compounding — the power of a power rule in compound growth.
    
5. Scaling in architecture or 3D printing: When you scale a 3D model by a factor of 2 in each dimension (length, width, height), the total volume scales by:
   
   \((2^1)^3 = 2^{1×3} = 2^3 = 8\)
   
   So the object becomes 8 times larger in volume, showing a real-world geometric use of the rule.