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

Answer 1

To simplify the expression \(3^7 \times 3^{-9}\) using the properties of integer exponents, we can use the property that states:

\[ a^m \times a^n = a^{m+n} \]

Applying this property here:

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

Next, we rewrite \(3^{-2}\) using the property of negative exponents, which states:

\[ a^{-n} = \frac{1}{a^n} \]

Thus, we have:

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

Now we calculate \(3^2\):

\[ 3^2 = 9 \]

Putting it all together, we get:

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

Therefore, the equivalent expression with only positive exponents is \(\frac{1}{9}\), and the final answer is:

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

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 simplify the expression \(\frac{2^2}{2^{-4}}\) using the properties of integer exponents, we can utilize the property of division:

\[ \frac{a^m}{a^n} = a^{m-n} \]

Applying this property to our expression:

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

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

\[ 2^6 = 64 \]

Therefore, the equivalent expression with only positive exponents is \(64\), and the final answer is:

\[ \boxed{64} \]

Answer 4

Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 times 5^-2/(-2)^-3

Answer 5

To simplify the expression \(4^0 \times \frac{5^{-2}}{(-2)^{-3}}\), we will use the properties of integer exponents.

Step 1: Simplify \(4^0\)

According to the property that any non-zero number raised to the power of 0 is equal to 1:

\[ 4^0 = 1 \]

Step 2: Simplify the fraction \(\frac{5^{-2}}{(-2)^{-3}}\)

Using the property of division of exponents:

\[ \frac{a^m}{b^n} = a^m \times b^{-n} \]

Thus, we can rewrite the expression as:

\[ \frac{5^{-2}}{(-2)^{-3}} = 5^{-2} \times (-2)^{3} \quad (\text{moving } (-2)^{-3} \text{ to the numerator}) \]

Next, calculate \((-2)^3\):

\[ (-2)^3 = -8 \]

Therefore, the fraction becomes:

\[ 5^{-2} \times (-8) \]

Step 3: Address the \(5^{-2}\)

Using the negative exponent property:

\[ 5^{-2} = \frac{1}{5^2} \]

Now we know that:

\[ 5^2 = 25 \]

So, now substituting:

\[ 5^{-2} = \frac{1}{25} \]

Step 4: Combine everything

Now combine the components:

\[ 4^0 \times \frac{5^{-2}}{(-2)^{-3}} = 1 \times \frac{-8}{25} = \frac{-8}{25} \]

Thus, the equivalent expression is:

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

Therefore, the simplified expression is:

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

Answer 6

with only positive exponents

Answer 7

To rewrite the expression \(4^0 \times \frac{5^{-2}}{(-2)^{-3}}\) with only positive exponents, we will go through the steps carefully again.

1. Start with \(4^0\): \[ 4^0 = 1 \] (Since any non-zero number raised to the power of zero equals 1).

2. Focus on \(\frac{5^{-2}}{(-2)^{-3}}\): This can be rewritten using the properties of exponents, specifically moving the negative exponent to the opposite part of the fraction: \[ \frac{5^{-2}}{(-2)^{-3}} = 5^{-2} \cdot (-2)^{3} \]

3. Calculate \((-2)^{3}\): \[ (-2)^{3} = -8 \] Thus, we can replace \((-2)^{-3}\) to get: \[ \frac{5^{-2}}{(-2)^{-3}} = 5^{-2} \cdot (-8) \]

4. Convert \(5^{-2}\) to positive exponent: \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

5. Combine the elements: Now we have: \[ 1 \cdot \frac{1}{25} \cdot (-8) = \frac{-8}{25} \]

6. Rewrite \( \frac{-8}{25} \) using positive exponents: We will express the negative numerator in a way that shows only positive exponents: \[ \frac{-8}{25} = -\frac{8}{25} \] (Note that the expression itself cannot be simplified further into positive exponent terms, as the \(-8\) prevents us from writing this result solely with positive exponents while holding onto the original negative value in standard mathematical form.)

Hence, the final expression including only positive exponents:

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

In case you want the expression purely as fractions with an indication of positivity, we say:

\[ \frac{8}{25} \text{ with a negative sign upfront} \]

Thus, the final answer is: \[ \boxed{-\frac{8}{25}} \]

This shows the stipulation of avoiding negative exponents while maintaining clarity of the negative value.

Answer 8

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

Answer 9

Let's simplify the expression \(4^0 \times \frac{5^{-2}}{(-2)^{-3}}\) step by step, ensuring we express the result using only positive exponents and no remaining exponent in the final answer.

Step 1: Simplify \(4^0\)

Using the property that any non-zero number raised to the power of 0 is equal to 1:

\[ 4^0 = 1 \]

Step 2: Simplify the fraction \(\frac{5^{-2}}{(-2)^{-3}}\)

Using the property of exponents for division:

\[ \frac{a^m}{b^n} = a^m \cdot b^{-n} \]

Applying this here gives:

\[ \frac{5^{-2}}{(-2)^{-3}} = 5^{-2} \cdot (-2)^{3} \]

Step 3: Calculate \((-2)^{3}\)

\[ (-2)^{3} = -8 \]

Step 4: Rewrite \(5^{-2}\)

Using the property for negative exponents:

\[ 5^{-2} = \frac{1}{5^2} \]

Since \(5^2 = 25\):

\[ 5^{-2} = \frac{1}{25} \]

Step 5: Combine everything

Now substitute back into the expression:

\[ 1 \times 5^{-2} \cdot (-2)^{3} = 1 \times \frac{1}{25} \cdot (-8) \]

This simplifies to:

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

Step 6: Express the result in positive exponent form

While \(-8\) is not a positive exponent, we can represent this fraction properly as is:

Thus, we can say:

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

However, since the problem requires a simplified fraction with no remaining exponents, we represent the result clearly as a fraction:

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

This final expression properly portrays the solution while adhering to the request for using only positive exponents in the calculations leading to the final fraction, which reflects no exponents left in the answer.

Answer 10

Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3 times 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 11

To simplify the expression \(\frac{(14^3 \times 27^{-7})^0}{3^4}\), we can follow these steps:

Step 1: Simplify \((14^3 \times 27^{-7})^0\)

Using the property that any non-zero number raised to the power of 0 is equal to 1, we have:

\[ (14^3 \times 27^{-7})^0 = 1 \]

Step 2: Substitute into the fraction

Now we substitute this result into the expression:

\[ \frac{1}{3^4} \]

Step 3: Simplify \(3^4\)

Now calculate \(3^4\):

\[ 3^4 = 81 \]

Step 4: Write the final result

Now we can rewrite the expression:

\[ \frac{1}{3^4} = \frac{1}{81} \]

Conclusion

The expression simplifies fully to:

\[ \boxed{\frac{1}{81}} \]

This represents a simplified fraction with no remaining exponents.