What are three possible ways to simplify (6^2)(2^4) and demonstrate why all three expressions are equivalent?
To simplify the expression (6^2)(2^4), we need to apply the rules of exponents. We can use three different methods to simplify the expression:
Method 1: Using the product of powers rule
According to the product of powers rule, when multiplying two exponential expressions with the same base, we can add their exponents. In this case, we have (6^2)(2^4), so we can add the exponents of 6 and 2 to get 6^2+4 = 6^6.
Method 2: Simplifying each exponential expression individually
We can first simplify both exponentials individually and then multiply them. 6^2 = 6 * 6 = 36, and 2^4 = 2 * 2 * 2 * 2 = 16. Hence, (6^2)(2^4) = 36 * 16.
Method 3: Using the power of products rule
According to the power of products rule, when raising a product to an exponent, we can distribute the exponent to each factor. In this case, we have (6^2)(2^4), so we can raise 6 to the power of 2 and 2 to the power of 4 separately, and then multiply the results.
6^2 = 6 * 6 = 36, and 2^4 = 2 * 2 * 2 * 2 = 16. Hence, (6^2)(2^4) = 36 * 16.
Now, let's show that all three expressions 6^6, 36 * 16, and (6^2)(2^4) are equivalent:
Using Method 1:
6^6 = 6^(2+4) = 6^2 * 6^4 = (6^2)(6^4)
6^6 = (6^2)(2^4) [Using the given expression]
Therefore, 6^6 = (6^2)(2^4) = 36 * 16.
Using Method 2:
(6^2)(2^4) = 36 * 16 [Given expression]
36 * 16 = 36 * 16
Therefore, (6^2)(2^4) = 36 * 16.
Using Method 3:
(6^2)(2^4) = 36 * 16 [Given expression]
36 * 16 = 36 * 16
Therefore, (6^2)(2^4) = 36 * 16.
In all three methods, we obtained the expression 6^6, 36 * 16, and (6^2)(2^4), which are all equivalent results.
To simplify the expression (6^2)(2^4), we can use the properties of exponents. Here are three possible ways to simplify it:
1. Using the product of powers property:
(6^2)(2^4) = 6^(2+4) = 6^6
2. Evaluating each exponent first and then multiplying:
6^2 = 6 * 6 = 36
2^4 = 2 * 2 * 2 * 2 = 16
So, (6^2)(2^4) = 36 * 16 = 576
3. Using the Commutative property:
(6^2)(2^4) = (2^4)(6^2)
Now, let's demonstrate the equivalence of all three expressions:
First, using the product of powers property:
(6^2)(2^4) = 6^(2+4) = 6^6
Next, evaluating each exponent separately:
6^2 = 6 * 6 = 36
2^4 = 2 * 2 * 2 * 2 = 16
So, (6^2)(2^4) = 36 * 16 = 576
Finally, using the Commutative property:
(6^2)(2^4) = (2^4)(6^2)
Therefore, all three expressions are equivalent since they all simplify to the same value.