Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7−3)7 ⋅ 7206−2=

Answer 1

Apply the product rule of integer exponents to find the numerical equivalent of   12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.(2 points

Answer 2

To apply the properties of integer exponents, we can use the following rules:

1. Power of a product: (a * b)^m = a^m * b^m
2. Power of a power: (a^m)^n = a^(m * n)
3. Negative exponent: a^-n = 1 / a^n

Now let's apply these rules to the expression (7^-3)7 * (720^6)^-2:

(7^-3)7 * (720^6)^-2
= (7^-3 * 7) * (720^6)^-2
= 7^-3 * 7 * (720^6)^-2
= 7^-3 * 7 * 720^-12

Next, let's use the rule of power of a power:

= (7 * 7) * (720^-12)^1

Now, let's use the rule of negative exponent:

= 49 * (1 / 720^12)

Therefore, the equivalent expression with only positive exponents is: 49 / 720^12.

To solve this expression, we need to simplify the numerator and the denominator separately. Let's start with the numerator:

49 = 7 * 7 = 49

Now let's simplify the denominator:

720^12 = (2^4 * 3^2 * 5)^12 = 2^(4*12) * 3^(2*12) * 5^12 = 2^48 * 3^24 * 5^12

Therefore, the final expression in simplified form is:

49 / (2^48 * 3^24 * 5^12)

This is the answer as an improper fraction.

Answer 3

To find the numerical equivalent of 12^-5 * 12^7, we can use the product rule of integer exponents.

The product rule of integer exponents states that when multiplying two bases with the same exponent, you can add the exponents.

So, in this case, we have:

12^-5 * 12^7

Using the product rule, we add the exponents:

= 12^(-5+7)

Simplifying the exponent:

= 12^2

Therefore, the numerical equivalent of 12^-5 * 12^7 is 12^2 which is equal to 144.

To further explain the use of the product rule in this example, we can break it down:

12^-5 can be written as 1/12^5 and 12^7 can be written as 12*12*12*12*12*12*12.

When multiplying these two expressions, we can add the exponents:

1/(12^5) * (12*12*12*12*12*12*12) = 1/12^5 * 12^7 = 12^(7-5) = 12^2 = 144.

So, by applying the product rule of integer exponents, we were able to simplify the expression and find the numerical equivalent.