Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7⋅3^−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
Apply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4 with only positive exponents, then solve the expression. Provide the solution as a whole number.
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a, a^0 = 1.
Using this property, we can rewrite the expression 4^0 as 1.
To rewrite 5^−2 with only positive exponents, we can use another property of integer exponents that states:
For any nonzero number a, a^(-n) = 1/a^n.
Applying this property, we can rewrite 5^−2 as 1/5^2.
To rewrite (−2)^−3 with only positive exponents, we can use the reciprocal property of integer exponents that states:
For any nonzero number a^(-n), 1/a^(-n) = a^n.
Applying this property, we can rewrite (−2)^−3 as 1/(-2)^3.
Now let's simplify the expression:
1 ⋅ (1/5^2) / (1/(-2)^3)
Simplifying further:
1 ⋅ 1/(5^2) ÷ 1/(-2)^3
Since division is the same as multiplying by the reciprocal of the denominator, we can rewrite the expression as:
1 ⋅ 1/(5^2) ⋅ ((-2)^3/1)
Simplifying further:
1/(5^2) ⋅ (-2)^3
Now let's simplify the exponents:
1/25 ⋅ (-8)
Finally, calculating the product:
1/25 ⋅ (-8) = -8/25
Therefore, the expression 4^0 ⋅ 5^−2/(−2)^−3 is equivalent to -8/25.
Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. For any nonzero number a, a^0 = 1.
2. For any nonzero numbers a and b, (a^m)^n = a^(m * n).
3. For any nonzero numbers a and b, (a * b)^n = a^n * b^n.
4. For any nonzero numbers a and b, (a / b)^n = a^n / b^n.
Now, let's apply these properties to the expression (14^3⋅27^−7)^0/3^4:
First, let's simplify the exponents within the parentheses using properties 1, 2, and 4:
(14^3⋅27^−7)^0 = (14^3)^0 * (27^−7)^0 = 1 * 1 = 1.
Next, let's simplify the denominator:
3^4 = 81.
Now, the expression becomes:
1/81.
Therefore, the expression (14^3⋅27^−7)^0/3^4 can be simplified to 1/81.
Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a and integers m and n, a^m ⋅ a^n = a^(m + n)
Using this property, we can rewrite the expression 3^7⋅3^−9 as:
3^(7 + (-9))
Simplifying the exponent:
3^(7 + (-9)) = 3^(-2)
To further simplify the expression with a negative exponent, we can use another property of integer exponents that states:
For any nonzero number a and integer n, a^(-n) = 1/a^n
Applying this property:
3^(-2) = 1/3^2
Simplifying the fraction:
1/3^2 = 1/9
Therefore, the expression 3^7⋅3^−9 is equivalent to 1/9.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a and integers m and n, a^m / a^n = a^(m - n).
Using this property, we can rewrite the expression 2^2/2^−4 as:
2^(2 - (-4))
Simplifying the exponent:
2^(2 - (-4)) = 2^(2 + 4)
Simplifying further:
2^(2 + 4) = 2^6
Therefore, the expression 2^2/2^−4 is equivalent to 2^6.
Solving the expression:
2^6 = 64
So, the solution to the expression 2^2/2^−4 is 64.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. For any nonzero number a, a^0 = 1.
2. For any nonzero numbers a and b, (a^m)^n = a^(m * n).
3. For any nonzero numbers a and b, (a * b)^n = a^n * b^n.
4. For any nonzero numbers a and b, (a / b)^n = a^n / b^n.
5. For any nonzero number a, a^(-n) = 1/a^n.
To simplify the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13, let's break it down step by step:
First, let's simplify the exponents within the parentheses using property 5:
(4^3 ⋅ 7^−2)^−5 = (4^3 ⋅ 1/7^2)^−5 = (4^3/7^2)^−5.
Now, let's simplify the denominator using property 4:
(4^3/7^2)^−5 = (4^3)^−5 / (7^2)^−5.
Now, let's simplify the exponents using property 2:
(4^3)^−5 / (7^2)^−5 = 4^(3 * -5) / 7^(2 * -5).
Simplifying further:
4^(3 * -5) = 4^(-15) = 1/4^15.
7^(2 * -5) = 7^(-10) = 1/7^10.
Now, let's substitute these values back into the expression:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = (1/4^15) / (1/7^10) ⋅ 4^(-13).
Applying properties 3 and 4 to simplify the expression further:
(1/4^15) / (1/7^10) ⋅ 4^(-13) = (1/4^15) ⋅ (7^10) / 4^13.
Using property 4 again to combine the fractions:
(1/4^15) ⋅ (7^10) / 4^13 = 7^10 / 4^(15 - 13).
Simplifying the exponent:
4^(15 - 13) = 4^2 = 16.
Now, substitute back into the expression:
7^10 / 16.
Therefore, the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 can be simplified to 7^10 / 16.