Question

Give a numerical example to represent this rule: For any nonnegative number a, any integer k and m , then: ᵏ√aᵐ= (ᵏ√a)ᵐ .

a. ³√2⁶ = (2^(1/3)⁶
b. ⁴√3⁸ = (3^(1/4)⁸
c. ⁵√5² = (5^(1/5)²
d. ²√7⁴ = (7^(1/2)⁴

Answer 1

The rule being illustrated is that taking a root of a power is equivalent to raising the root to that power, which can be shown with the provided numerical examples, demonstrating the power of a root equals the root of a power for any nonnegative number a, and integers k and m.

Step-by-step explanation:

The question involves the rule that for any nonnegative number a, any integer k, and m, then k√am = (k√a)m. This rule shows that the power of a root is equal to the root of a power. Using the rule that when you raise a power to another power, you can multiply the exponents, we'll demonstrate with the provided numerical examples:

• For a = 2, k = 3, and m = 6: ³√26 = (21/3)6
• For a = 3, k = 4, and m = 8: ⁴√38 = (31/4)8
• For a = 5, k = 5, and m = 2: ⁵√52 = (51/5)2
• For a = 7, k = 2, and m = 4: ²√74 = (71/2)4

These examples help illustrate that raising a root to an integer power is equivalent to raising the number itself to the power of the result of dividing the integer by the root.