Simplifying Exponential Expressions Using the Power Rule

Key Concept: Power Rule of Exponents

• Power Rule: If you have (x^a) raised to the (b)-th power, it's equal to (x^{a \times b}).
• Example: ((x^2)^3 = x^{2\times3} = x^6)
  • This means multiplying (x^2) three times: (x^2 \times x^2 \times x^2).
  • Using the property of adding exponents for like bases: (x^{2+2+2} = x^6).

Simplifying the Given Expression

• Original expression: (((x^3)^4 \times x^2)^5)

Step-by-Step Simplification

1. Apply the Power Rule:

   • ((x^3)^4 = x^{3 \times 4} = x^{12})
   • ((x^2)^5 = x^{2 \times 5} = x^{10})

2. Combine Like Terms:

   • Multiply (x^{12}) by (x^{10}) using the rule for multiplying powers with the same base:
   • (x^{12} \times x^{10} = x^{12 + 10} = x^{22})

Final Result

• The expression (((x^3)^4 \times x^2)^5) simplifies to (x^{22}).

Important Notes

• Remember to apply the power rule to simplify expressions effectively.
• When multiplying like bases, you add the exponents.