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 Integer Exponents? (1 point)
(7 ^ 2)/((- 4) ^ 3)
(4 ^ 3)/(7 ^ 2)
((- 4) ^ 3)/(7 ^ 3)
1/(7 ^ 2 * (- 4) ^ - 2)
The correct answer is (7 ^ 2)/((- 4) ^ 3).
Which of the following is the equivalent expression to (15^-3x4^7)^0/4^-3 that has been generated by applying the Properties of
Integer Exponents? (1 point)
The equivalent expression to (15^-3x4^7)^0/4^-3 that has been generated by applying the Properties of Integer Exponents is 1/4^(-3).
To find an equivalent expression with positive exponents, we can use the properties of integer exponents:
1. For any non-zero number a, a^0 = 1.
2. For any non-zero number a, a^(-n) = 1/a^n
Applying these properties, let's simplify the given expression step by step:
(15 ^ 0 * 7 ^ - 2)/((- 4) ^ - 3)
Step 1: Simplify inside the parentheses:
((15 ^ 0) * (7 ^ -2)) / ((-4) ^ -3)
Step 2: Apply the property a^0 = 1:
(1 * (7 ^ -2)) / ((-4) ^ -3)
Step 3: Apply the property a^(-n) = 1/a^n:
(1/(7^2)) / (1/(-4)^3)
Step 4: Simplify the expression:
1/(7^2 * (-4)^3)
Therefore, the equivalent expression with positive exponents that has been generated by applying the properties of integer exponents is:
1/(7^2 * (-4)^3)
To find the equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the given expression step by step:
(15 ^ 0 * 7 ^ - 2)/((- 4) ^ - 3)
Any number raised to the power of 0 is equal to 1, so we can simplify (15 ^ 0) to 1:
(1 * 7 ^ - 2)/((- 4) ^ - 3)
Next, we can use the rule that a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent. In this case, 7 ^ -2 can be rewritten as 1/(7 ^ 2), and (-4) ^ -3 can be rewritten as 1/((-4) ^ 3):
(1/(7 ^ 2))/ (1/((- 4) ^ 3))
Now, dividing by a fraction is the same as multiplying by the reciprocal, so we can simplify further:
1/(7 ^ 2) * ((- 4) ^ 3/1)
Since any number raised to the power of 1 is the number itself, we can remove the exponent of 1:
1/(7 ^ 2) * (-4) ^ 3
Now, to rewrite this expression with only positive exponents, we can use the property that the reciprocal of a number raised to a positive exponent can be rewritten as the number itself raised to a negative exponent. In this case, 1/(7 ^ 2) can be rewritten as 7 ^ -2, so we get:
(7 ^ -2) * (-4) ^ 3
Finally, using the commutative property of multiplication, we can rearrange the terms:
(-4) ^ 3 * (7 ^ -2)
So the equivalent expression with only positive exponents that has been generated by applying the properties of integer exponents is:
(7 ^ 2)/((-4) ^ 3)