Math Problem Statement

(4^5)^6 × (4^6)^5

Solution

The mathematical expression provided in the image is:

(45)6×(46)5(4^5)^6 \times (4^6)^5

Step-by-step solution:

  1. Apply the Power Rule: (am)n=amn(a^m)^n = a^{m \cdot n}

    • For (45)6(4^5)^6, this becomes 456=4304^{5 \cdot 6} = 4^{30}.
    • For (46)5(4^6)^5, this becomes 465=4304^{6 \cdot 5} = 4^{30}.
  2. Combine the Exponents:

    • The base 44 is the same, so use the rule aman=am+na^m \cdot a^n = a^{m+n}.
    • 430×430=430+30=4604^{30} \times 4^{30} = 4^{30 + 30} = 4^{60}.

Final Answer:

4604^{60}

Would you like further explanation, or to evaluate 4604^{60} numerically? Let me know!


Related Questions:

  1. What are the basic laws of exponents, and how are they applied here?
  2. How do you simplify expressions with the same base in multiplication or division?
  3. What is the numerical value of 4604^{60}?
  4. How do exponents behave when fractions or negatives are involved?
  5. What is the general rule for simplifying ((am)n)p((a^m)^n)^p?

Tip:

Always simplify powers step-by-step to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Exponents
Power Rule
Multiplication of Exponents

Formulas

(a^m)^n = a^(m*n)
a^m * a^n = a^(m+n)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10