which of the following is an equivalent expression to 5^9 . 5^-13 with only positive exponents that has been generated by applying the properties of integer exponents?
a 1/5^4
b 1/625
c 5^4
d 5^9/5^-13
To simplify the expression 5^9 . 5^-13 with positive exponents, we can use the property of exponents that states: a^m . a^n = a^(m + n).
In this case, 5^9 . 5^-13 can be simplified as 5^(9 + (-13)) = 5^(-4).
To express 5^(-4) with a positive exponent, we can use another property of exponents that states: a^-n = 1/a^n.
Therefore, 5^(-4) can be rewritten as 1/5^4.
The equivalent expression with positive exponents is:
a) 1/5^4
which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9?
a only the Property of Negative Exponents
b only the power rule of exponents
c only the product rule of exponents
d the property of negative exponents and the product rule of exponents
To generate the equivalent expression 3^14 from 3^5/3^-9, we can use the property of exponents, specifically the property of negative exponents and the product rule of exponents.
First, using the property of negative exponents, we know that a^-n = 1/a^n. So, we can rewrite 3^-9 as 1/3^9.
Next, using the product rule of exponents, a^m/a^n = a^(m-n). In this case, we have 3^5/3^9 = 3^(5-9) = 3^-4.
Combining the two steps, we have 3^5/3^-9 = 3^5/(1/3^9) = 3^5 * 3^9 = 3^(5+9) = 3^14.
Therefore, the property of negative exponents and the product rule of exponents were used to generate the equivalent expression 3^14 from 3^5/3^-9.
d) the property of negative exponents and the product rule of exponents
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?
a (-4)^3/7^2
b 7^2/(-4)^3
c 4^3/7^2
d 1/7^2 . (-4)^-3
To simplify the expression 15^0 . 7^-2 / (-4)^-3 with only positive exponents, we can use the property of exponents that states: a^0 = 1 (for a ≠ 0) and the property that a^-n = 1/a^n.
First, let's simplify 15^0. According to the property a^0 = 1 (for a ≠ 0), we know that 15^0 is equal to 1.
Next, we can apply the property of negative exponents to simplify 7^-2 and (-4)^-3. Using the property a^-n = 1/a^n, we can rewrite 7^-2 as 1/7^2 and (-4)^-3 as 1/(-4)^3.
Combining these simplifications, we have:
15^0 . 7^-2 / (-4)^-3 = 1 . 1/7^2 / 1/(-4)^3
Now, let's simplify this expression using the property of division, a/b ÷ c/d = a/b * d/c:
1 . 1/7^2 / 1/(-4)^3 = 1 * (-4)^3 / (1/7^2)
Finally, we can simplify this expression further:
1 * (-4)^3 / (1/7^2) = (-4)^3 * (7^2/1)
Therefore, an equivalent expression to 15^0 . 7^-2 / (-4)^-3 with only positive exponents is:
c) 4^3/7^2
which of the following is the equivalent expression to (15^-3 . 4^7)^0/4^-3 that has been generated by applying the Properties of Integer Exponents?
a 64
b 1 . 4^-3
c 1/64
d 60/4^-3
To simplify the expression (15^-3 . 4^7)^0 / 4^-3 using the properties of integer exponents, we need to understand a few rules.
First, any number raised to the power of 0 is equal to 1.
Second, when we have a negative exponent, it can be rewritten as the reciprocal of the positive exponent. That is, a^-n = 1/a^n.
Now, let's simplify the expression step by step:
(15^-3 . 4^7)^0 / 4^-3 = 1 / 4^-3
Next, using the rule that a^-n = 1/a^n, we can rewrite 4^-3 as 1/4^3:
= 1 / 1/4^3
This is equivalent to multiplying by the reciprocal, so we have:
= 1 * 4^3
Finally, we can simplify the expression further:
= 4^3
= 4 * 4 * 4
= 64
Therefore, the equivalent expression to (15^-3 . 4^7)^0 / 4^-3, using the properties of integer exponents, is:
a) 64