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The Power of a Power Rule makes exponentiation simpler. This rule states that if you have an exponent raised to another exponent, you can multiply those exponents together. Mathematically, this is represented as: \[ (a^m)^n = a^{mn} \]
This rule is useful when dealing with complicated expressions. Instead of repeatedly applying exponents step-by-step, you can directly simplify by multiplying them. For example, given \[ (s^{\frac{3}{7}})^{\frac{1}{6}} \], you can simplify it by multiplying the exponents resulting in \[ s^{\frac{3}{7} \cdot \frac{1}{6}} = s^{\frac{3}{42}} = s^{\frac{1}{14}} \].
In another example, \[ (m^{\frac{4}{3}})^{\frac{3}{4}} \] simplifies to \[ m^{\frac{4}{3} \cdot \frac{3}{4}} = m^1 = m \]. This method is vital for handling exponential expressions quickly.

Simplifying expressions using the power of a power rule involves breaking down complex terms into simpler forms.
Let's take the example \[ \left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}} \].
The first step is to apply the power rule separately to the base 64 and the variable term \[ s^{\frac{3}{7}} \].
For the constant base 64, simplify it as follows:

• Recognize that 64 is \[ 2^6 \]
• Thus \[ (2^6)^{\frac{1}{6}} = 2 \]

For the variable part \[ (s^{\frac{3}{7}})^{\frac{1}{6}} \], multiply the exponents:

• Calculate \[ \frac{3}{7} \cdot \frac{1}{6} = \frac{3}{42} = \frac{1}{14} \]
• Thus \[ s^{\frac{1}{14}} \]

Combining both simplified parts results in:
\[ 2 s^{\frac{1}{14}} \]following these steps makes the simplification straightforward.

Exponents indicate how many times a number, called the base, is multiplied by itself. For example, \[ 2^3 = 2 \times 2 \times 2 = 8 \]. Exponents follow specific rules which make complex calculations simpler. Here are the key rules:

• Product Rule: \[ a^m \cdot a^n = a^{m+n} \]
• Quotient Rule: \[ \frac{a^m}{a^n} = a^{m-n} \]
• Power of a Power Rule: \[ (a^m)^n = a^{mn} \]
• Zero Exponent Rule: \[ a^0 = 1 \] if \[ a e 0 \]
• Negative Exponent Rule: \[ a^{-m} = \frac{1}{a^m} \]

Using these rules, let’s simplify \[ \left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}} \]:

• Apply the power of a power rule individually:

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• For \[ m^{\frac{4}{3}} \], multiply exponents: \[ \frac{4}{3} \cdot \frac{3}{4} = 1 \]. Hence \[ m^1 = m \]
• For \[ n^{\frac{1}{2}} \], multiply: \[ \frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8} \]. Hence \[ n^{\frac{3}{8}} \]

Combine both results to get:
\[ m n^{\frac{3}{8}} \]Mastering these exponent rules is crucial for efficient problem-solving in algebra.