Math Problem Statement

(2³)³ ÷ (2⁴)⁴

Solution

Let's solve the expression (23)3÷(24)4(2^3)^3 \div (2^4)^4.

  1. Simplify (23)3(2^3)^3: (23)3=23×3=29(2^3)^3 = 2^{3 \times 3} = 2^9

  2. Simplify (24)4(2^4)^4: (24)4=24×4=216(2^4)^4 = 2^{4 \times 4} = 2^{16}

  3. Now, the expression becomes: 29216\frac{2^9}{2^{16}}

  4. Apply the quotient rule for exponents, aman=amn\frac{a^m}{a^n} = a^{m-n}: 2916=272^{9-16} = 2^{-7}

  5. Finally, 272^{-7} can also be written as: 127=1128\frac{1}{2^7} = \frac{1}{128}

So, (23)3÷(24)4=1128(2^3)^3 \div (2^4)^4 = \frac{1}{128}.

Would you like more details on any step? Here are some related questions:

  1. How do you simplify other exponent expressions like (32)4(3^2)^4?
  2. What is the rule for dividing expressions with the same base?
  3. How can negative exponents be converted to fractions?
  4. How do you handle exponents with different bases?
  5. Can you explain the power rule of exponents in more detail?

Tip: When multiplying or dividing expressions with the same base, always use the exponent rules to simplify the calculation.

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Math Problem Analysis

Mathematical Concepts

Exponents
Quotient Rule
Negative Exponents

Formulas

Power of a Power Rule: (a^m)^n = a^{m*n}
Quotient Rule for Exponents: a^m ÷ a^n = a^{m-n}
Negative Exponents: a^{-n} = 1/a^n

Theorems

Exponent Rules

Suitable Grade Level

Grades 8-10