Math Problem Statement

Simplify the expression: (8^5)^6 · 8^2.

Solution

The given expression in the image is:

(85)682\left(8^5\right)^6 \cdot 8^2

Step-by-Step Simplification:

  1. Apply the Power of a Power Rule: The rule for powers of powers is (am)n=amn\left(a^m\right)^n = a^{m \cdot n}. So, (85)6=856=830\left(8^5\right)^6 = 8^{5 \cdot 6} = 8^{30}

  2. Combine Powers with the Same Base: When multiplying powers with the same base, the exponents are added: 83082=830+2=8328^{30} \cdot 8^2 = 8^{30 + 2} = 8^{32}

Thus, the simplified expression is:

8328^{32}

Would you like more details on any part of this explanation?

Related Questions:

  1. What is the power of a product rule in exponents?
  2. How can you simplify (34)235(3^4)^2 \cdot 3^5?
  3. What is the expanded form of 8328^{32}?
  4. How do you handle negative exponents when simplifying powers?
  5. Can you explain the zero exponent rule?

Tip: Remember, when multiplying powers of the same base, always add the exponents!

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Math Problem Analysis

Mathematical Concepts

Exponents
Power of a Power Rule
Multiplying Powers

Formulas

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

Theorems

Power of a Power Rule
Product of Powers Property

Suitable Grade Level

Grades 8-10