Can you provide three distinct mathematical expressions to represent the product of two powers, using any numbers you prefer? Justify how all of these expressions are equivalent.
Certainly! To represent the product of two powers, we can use the following three mathematical expressions:
1. First Expression: a^m * a^n
2. Second Expression: a^(m + n)
3. Third Expression: a^(m) * a^(n)
Let's see how all of these expressions are equivalent using the example values a = 2, m = 3, and n = 4.
1. First Expression: a^m * a^n = 2^3 * 2^4 = (2 * 2 * 2) * (2 * 2 * 2 * 2) = 8 * 16 = 128
2. Second Expression: a^(m + n) = 2^(3 + 4) = 2^7 = 128
3. Third Expression: a^(m) * a^(n) = 2^3 * 2^4 = (2 * 2 * 2) * (2 * 2 * 2 * 2) = 8 * 16 = 128
As you can see, all three expressions yield the same result, which is 128. This demonstrates that they are mathematically equivalent.
Sure! Let's take two powers:
1. Expression 1: a^m * a^n
2. Expression 2: a^(m + n)
3. Expression 3: a^(m*n)
In all three expressions, 'a' represents the same number being raised to different powers 'm' and 'n'.
To justify their equivalence:
Expression 1: We know that when multiplying numbers with the same base, we can add their exponents. So, a^m * a^n = a^(m + n).
Expression 2: This follows the same principle as Expression 1. By multiplying two powers with the same base 'a', we can combine their exponents to simplify as a single power, a^(m + n).
Expression 3: This expression represents the property of exponentiation, where raising a power to another power results in multiplying the exponents. In this case, a^m raised to the power of n becomes a^(m * n).
Thus, all three expressions are equivalent representations of the product of two powers.