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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.