Apply the Properties of Integer Exponents to generate equivalent expressions to 3 to the power of 7 * 3 to the power of −9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate an equivalent expression with only positive exponents, we can use the property that says:
a^(m) * a^(n) = a^(m+n).
Therefore, we can rewrite 3^7 * 3^(-9) as 3^(7 + -9).
Simplifying, 7 + -9 = -2.
So, 3^7 * 3^(-9) is equivalent to 3^(-2).
To solve 3^(-2), we can use the property that says:
a^(-n) = 1 / a^(n).
Therefore, 3^(-2) = 1 / 3^(2).
Simplifying, 3^(2) = 3 * 3 = 9.
So, 3^(-2) = 1 / 9.
Therefore, the expression 3^7 * 3^(-9) simplifies to 1/9.
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.
To generate an equivalent expression with only positive exponents, we can use the property that says:
a^(m) / a^(n) = a^(m-n).
Therefore, we can rewrite 2^2 / 2^(-4) as 2^(2 - (-4)).
Simplifying, 2 - (-4) = 2 + 4 = 6.
So, 2^2 / 2^(-4) is equivalent to 2^(6).
To solve 2^(6), we simply calculate 2^6:
2^6 = 2 * 2 * 2 * 2 * 2 * 2 = 64
Therefore, the expression 2^2 / 2^(-4) simplifies to 64.
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 an equivalent expression with only positive exponents, we can use the property that says:
a^(m) / a^(n) = a^(m-n)
Applying this property, we can rewrite the expression 4^0 * 5^(-2) / (-2)^(-3) as:
1 * (1 / 5^2) / (1 / (-2)^3)
Simplifying further, we have:
1 * (1 / 25) / (1 / (-8))
To divide a fraction by another fraction, we can multiply by the reciprocal of the divisor. This means multiplying the numerator by the reciprocal of the denominator. Performing this multiplication, we get:
1 * (1 / 25) * (-8 / 1)
Simplifying, we have:
-8 / 25
Therefore, the expression 4^0 * 5^(-2) / (-2)^(-3) simplifies 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 an equivalent expression with only positive exponents, we can use the property that says:
(a^m)^n = a^(m*n)
Therefore, we can rewrite (14^3 * 27^(-7))^0 as 14^(3*0) * 27^(-7*0).
Simplifying further, we have 14^0 * 27^0.
Any number raised to the power of 0 is always equal to 1. Therefore, 14^0 and 27^0 are both equal to 1.
So, (14^3 * 27^(-7))^0 simplifies to 1.
To solve the expression 1 / 3^4, we can use the fact that any number raised to the power of 0 is equal to 1, so we have 1 / 81.
Therefore, the expression (14^3 * 27^(-7))^0 / 3^4 simplifies 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 an equivalent expression with only positive exponents, we can use the property that says:
(a^m)^n = a^(m*n)
Therefore, we can rewrite (4^3 * 7^(-2))^(-5) as 4^(3*(-5)) * 7^((-2)*(-5)).
Simplifying further, we have 4^(-15) * 7^10.
Now, to divide two numbers with the same base, but different exponents, we can subtract the exponents. Therefore, we can rewrite 7^9 / 7^10 as 7^(9-10).
Simplifying further 7^(9-10) equals to 7^-1.
Thus, the expression (4^3 * 7^(-2))^(-5) / 7^9 simplifies to 4^(-15) * 7^-1.
Now, to further simplify 4^(-15) * 7^-1, we can rewrite it as 1 / 4^15 * 1 / 7.
Since the reciprocal of a number raised to a certain power is that number raised to the negative of that power, we can rewrite the expression as 1 / (4^15 * 7).
Now, to express the expression (1 / (4^15 * 7)) * 4^(-13), we perform the multiplication:
(1 / (4^15 * 7)) * 4^(-13) = 1 / ((4^15 * 7) / (4^13)).
Now, to divide two numbers with the same base, we subtract the exponents. Therefore, (4^15 * 7) / (4^13) simplifies to 4^(15-13) * 7.
Simplifying further, we have 4^2 * 7, which equals to 16 * 7 = 112.
So, the expression (4^3 * 7^(-2))^(-5) / 7^9 * 4^(-13) simplifies to 112.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
The property we can use here is the product of powers property:
a^m * a^n = a^(m+n)
Using this property, we can rewrite the expression 3^7 * 3^(-9).
By applying the property, we add the exponents:
3^7 * 3^(-9) = 3^(7+(-9))
Simplifying the exponent, we have:
3^(-2)
To simplify this further, we can use the property of negative exponent:
a^(-n) = 1 / a^n
Applying this property to our expression, we get:
3^(-2) = 1 / 3^2
Simplifying further, we have:
1 / 3^2 = 1 / 9
So, the final simplified fraction with no remaining exponents is 1/9.