Math Problem Statement

Simplify the expression (2m5)6 .

Solution

To simplify the expression (2m5)6(2m^5)^6, follow these steps:

Step 1: Apply the Power of a Product Rule

The power of a product rule states that (ab)n=anbn(ab)^n = a^n \cdot b^n. In this case, you can apply it to both the constant 22 and the variable term m5m^5:

(2m5)6=26(m5)6(2m^5)^6 = 2^6 \cdot (m^5)^6

Step 2: Simplify Each Term

  • 26=642^6 = 64
  • For (m5)6(m^5)^6, use the power of a power rule, which states (xa)b=xab(x^a)^b = x^{a \cdot b}. So, (m5)6=m56=m30(m^5)^6 = m^{5 \cdot 6} = m^{30}.

Step 3: Write the Final Expression

Now combine everything:

(2m5)6=64m30(2m^5)^6 = 64m^{30}

Thus, the simplified expression is:

64m30\boxed{64m^{30}}

Would you like more details or have any questions?


Here are 5 related questions to expand on this:

  1. How would you simplify (3x2)4(3x^2)^4?
  2. What happens when simplifying (5y3)2(5y^3)^2?
  3. How does the power rule work when applying it to a negative exponent, e.g., (2m3)4(2m^{-3})^4?
  4. How do you simplify a product of two powers, e.g., (2m2)3(4m4)2(2m^2)^3 \cdot (4m^4)^2?
  5. Can you explain how the product rule is applied to expressions with multiple terms?

Tip: When raising a product to a power, apply the exponent to each factor inside the parentheses.

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Math Problem Analysis

Mathematical Concepts

Exponents
Power of a Product Rule
Power of a Power Rule

Formulas

(ab)^n = a^n * b^n
(x^a)^b = x^{a * b}

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10