Question

Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7 x 3^-9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

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 2

Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 x 5^-2/(-2)^-3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 3

To solve the expression \(3^7 \times 3^{-9}\) using the properties of integer exponents, we will start by applying the Product of Powers Property, which states that when multiplying two expressions with the same base, we add the exponents:

\[
3^7 \times 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2}
\]

Next, we need to convert the expression \(3^{-2}\) to use only positive exponents. We can do this by using the property that \(a^{-n} = \frac{1}{a^n}\):

\[
3^{-2} = \frac{1}{3^2}
\]

Now, we compute \(3^2\):

\[
3^2 = 9
\]

Thus, we can rewrite \(3^{-2}\) as:

\[
3^{-2} = \frac{1}{9}
\]

Consequently, the equivalent expression for \(3^7 \times 3^{-9}\) with only positive exponents is:

\[
\boxed{\frac{1}{9}}
\]

Answer 4

To simplify the expression \(\frac{2^2}{2^{-4}}\) using the properties of integer exponents, we can apply the Quotient of Powers Property, which states that when dividing two expressions with the same base, we subtract the exponents:

\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]

Now we can calculate \(2^6\):

\[
2^6 = 64
\]

Thus, the equivalent expression for \(\frac{2^2}{2^{-4}}\) with only positive exponents is:

\[
\boxed{64}
\]

Answer 5

We start with the expression:

\[
4^0 \times 5^{-2} \div (-2)^{-3}
\]

### Step 1: Simplifying with Properties of Exponents

1. **Calculate \(4^0\)**:
\[
4^0 = 1
\]
Therefore, we can simplify the expression to:
\[
1 \times 5^{-2} \div (-2)^{-3} = 5^{-2} \div (-2)^{-3}
\]

2. **Rewrite the division of exponents as multiplication**:
To divide by \((-2)^{-3}\), we can use the property that \(\frac{a^m}{b^n} = a^m \times b^{-n}\):
\[
5^{-2} \times (-2)^{3}
\]

### Step 2: Rewrite with Positive Exponents

Now we can rewrite the expression:
\[
\frac{(-2)^3}{5^2}
\]

### Step 3: Simplifying the Expression

1. **Calculate \((-2)^3\)**:
\[
(-2)^3 = -8
\]

2. **Calculate \(5^2\)**:
\[
5^2 = 25
\]

Now, we can substitute these values into the expression:
\[
\frac{-8}{25}
\]

Thus, the final result, expressed as a simplified fraction with no remaining exponents, is:

\[
\boxed{\frac{-8}{25}}
\]