Which of the following is an equivalent expression to 15^0•7^-2/(-4)^-3 with only positive exponents that has been generated by applying the properties of intiger exponents?
1/7^2•(-4)^-2
7^2/(-4)^3
4^3/7^2
(-4)^3/7^2
To find an equivalent expression with only positive exponents, we can use the rule that a negative exponent is equivalent to the reciprocal of the positive exponent.
Starting with the expression 15^0•7^-2/(-4)^-3, we can rewrite 7^-2 as 1/7^2 and (-4)^-3 as 1/(-4)^3.
So the equivalent expression is: 15^0 • 1/7^2 • 1/(-4)^3
Which simplifies to: 1 • 1/49 • 1/(-64)
= 1/49 • 1/(-64)
= 1/((-64)(49))
= (-4)^3/7^2
Therefore, the correct equivalent expression with only positive exponents is (-4)^3/7^2.
Option D is the correct answer.
To simplify the expression and generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
Let's break down the given expression step by step:
15^0 • 7^-2 / (-4)^-3
First, let's simplify the exponents.
Any number with an exponent of 0 is equal to 1, so 15^0 = 1.
Using the property of negative exponents, any non-zero number raised to a negative exponent becomes the reciprocal of the number raised to the positive exponent. Therefore, 7^-2 = 1/7^2.
Similarly, (-4)^-3 = 1/(-4)^3.
Substituting these values back into the expression:
1 • (1/7^2) / (1/(-4)^3)
Next, let's multiply the numerators and denominators:
1/7^2 • (-4)^3
To simplify, we can multiply the numbers with the same base and add the exponents:
1/7^2 • (-4^3)
Simplifying the exponent:
1/7^2 • (-64)
To make all the exponents positive, we can rewrite -64 as (-1)^3 • 4^3:
1/7^2 • ((-1)^3 • 4^3)
Now, let's simplify each term:
1/7^2 = 1/49
(-1)^3 = -1 (since an odd power of -1 is always -1)
4^3 = 64
Substituting these values back into the expression:
(1/49) • (-1 • 64)
Since -1 • 64 is equal to -64, we have:
(1/49) • (-64)
Finally, the equivalent expression with only positive exponents generated by applying the properties of integer exponents is:
-64/49
So, the correct choice is not among the given options.
To find the equivalent expression with only positive exponents, we can apply the properties of integer exponents.
Let's break down the given expression step by step:
15^0 • 7^-2 / (-4)^-3
First, we know that any number raised to the power of 0 is equal to 1. So, 15^0 simplifies to 1:
1 • 7^-2 / (-4)^-3
Next, applying the property of negative exponents - when a number has a negative exponent, it can be moved to the denominator (or vice versa) and the exponent becomes positive.
So, moving 7^-2 to the denominator with a positive exponent, we get:
1 / (7^2) / (-4)^-3
Now, let's simplify (-4)^-3. A negative exponent on a base can be rewritten as the reciprocal of the base raised to the positive exponent. So, (-4)^-3 can be transformed into 1/(-4)^3:
1 / (7^2) / 1/(-4)^3
Simplifying further, dividing by a fraction is the same as multiplying by its reciprocal. So, dividing by 1/(-4)^3 is the same as multiplying by (-4)^3:
1 / (7^2) * (-4)^3
Finally, we can rearrange the expression by multiplying the numerators together and the denominators together:
1 * (-4)^3 / (7^2)
The equivalent expression with only positive exponents and generated by applying the properties of integer exponents is:
(-4)^3 / 7^2