Math Problem Statement

Solve each problem using the laws of exponents.

Solution

The worksheet you uploaded involves solving problems using the laws of exponents. Here’s a detailed breakdown of each question along with the solutions:

  1. (12)4\left( \frac{1}{2} \right)^4

    • Solution: (12)4=1424=116\left( \frac{1}{2} \right)^4 = \frac{1^4}{2^4} = \frac{1}{16}
  2. 313^1

    • Solution: 31=33^1 = 3
  3. 32×333^2 \times 3^3

    • Solution: Using the law am×an=am+na^m \times a^n = a^{m+n}, we get 35=2433^5 = 243
  4. 34×323^4 \times 3^2

    • Solution: Using the same law am×an=am+na^m \times a^n = a^{m+n}, we get 36=7293^6 = 729
  5. (22)2(2^2)^2

    • Solution: Using the power of a power rule (am)n=am×n(a^m)^n = a^{m \times n}, we get 24=162^4 = 16
  6. 202^0

    • Solution: Any number raised to the power of 0 equals 1, so 20=12^0 = 1
  7. 32×323^2 \times 3^2

    • Solution: Using the multiplication rule for exponents, 32×32=34=813^2 \times 3^2 = 3^4 = 81
  8. (2×3)3(2 \times 3)^3

    • Solution: Apply the exponent to both numbers, 23×33=8×27=2162^3 \times 3^3 = 8 \times 27 = 216
  9. 343^4

    • Solution: 34=813^4 = 81
  10. 33×323^3 \times 3^2

  • Solution: Using the same exponent rule, 35=2433^5 = 243

Would you like any further details or clarifications on these solutions?

Related Questions:

  1. What is the general rule for multiplying powers with the same base?
  2. How does the zero exponent rule apply to different bases?
  3. Can you explain the power of a product rule with a different example?
  4. What happens if you multiply two powers with different bases but the same exponent?
  5. What is the result of ana^{-n}, and how does it relate to positive exponents?

Tip:

When multiplying powers with the same base, remember to add the exponents, and when raising a power to another power, you multiply the exponents.

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Math Problem Analysis

Mathematical Concepts

Exponents
Laws of Exponents

Formulas

Product of Powers: a^m × a^n = a^(m+n)
Power of a Power: (a^m)^n = a^(m×n)
Zero Exponent Rule: a^0 = 1
Power of a Product: (ab)^n = a^n × b^n

Theorems

Laws of Exponents

Suitable Grade Level

Grade 6-8