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.

Question

Pls explain simply

Answer 1

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

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

In this case, we let \(a = 3\), \(m = 7\), and \(n = -9\):

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

Now, since we want to write this with only positive exponents, we can apply the negative exponent rule, which states:

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

Using this on \(3^{-2}\):

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

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

\[ 3^2 = 9 \]

Thus, we can rewrite our expression as:

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

So the equivalent expression of \(3^7 \times 3^{-9}\) with only positive exponents is \(\frac{1}{9}\).

Therefore, the final answer is:

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

Answer 2

What's /[
\frac{1}{9}
\]???
Idk what it means

Answer 3

I apologize for the confusion! It looks like there was an unintended formatting issue in my response.

The expression \(\frac{1}{9}\) is simply a mathematical way to express "one ninth." It means you have a fraction where 1 is the numerator (the top number) and 9 is the denominator (the bottom number).

To clarify, the final answer to your original question is:

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

This means that after applying the properties of integer exponents to simplify \(3^7 \times 3^{-9}\), we find that the result is \(\frac{1}{9}\). If you have any more questions or need further clarification, feel free to ask!

Answer 4

Does frac stand for fraction and what are the 1 and 9, are they exponents?

Answer 5

Yes, "frac" is short for "fraction," and in mathematical notation, it is used to represent a fraction.

In the expression \(\frac{1}{9}\):

• The 1 is the numerator (the top part of the fraction).
• The 9 is the denominator (the bottom part of the fraction).

Neither the 1 nor the 9 are exponents; they are simply the numbers that make up the fraction. This means that \(\frac{1}{9}\) represents one part out of nine equal parts.

So, to clarify the components:

• \(\frac{1}{9}\) means "one divided by nine," which is a simple fraction.
• There are no exponents involved in this expression.

If you need more information or have other questions, just let me know!

Answer 6

Ok that was correct (1 over 9)
Now, apply the property of integer exponents to generate equivalent expressions to 2^2 over 2^-4 with only positive exponents. solve the expression and provide the solution as a whole number

Answer 7

To solve the expression \(\frac{2^2}{2^{-4}}\) using the properties of integer exponents, we can use the property that states:

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

In this case, we have:

• \(a = 2\)
• \(m = 2\)
• \(n = -4\)

Now applying the property:

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

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

\[ 2^6 = 64 \]

Thus, the expression \(\frac{2^2}{2^{-4}}\) simplifies to \(64\).

The final answer is:

\[ 64 \]

Answer 8

Correct.
Next, apply the properties of integer exponentd to 4^0 times 5^-2 over (-2)^-3 with only positive exponents. Your answer will be a simplified fraction with no remaining exponents

Answer 9

To simplify the expression \(\frac{4^0 \times 5^{-2}}{(-2)^{-3}}\) using the properties of integer exponents, we will break it down step by step.

1. Evaluating \(4^0\):

   • According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. So, we have: \[ 4^0 = 1 \]

2. Substituting 4^0 into the expression:

   • This allows us to simplify our original expression: \[ \frac{1 \times 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}} \]

3. Using the property of negative exponents:

   • Recall that \(a^{-n} = \frac{1}{a^n}\). This allows us to rewrite the expression: \[ 5^{-2} = \frac{1}{5^2} \] \[ (-2)^{-3} = \frac{1}{(-2)^3} \] Using these, we can update our expression: \[ \frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{5^2}}{\frac{1}{(-2)^3}} \]

4. Dividing fractions:

   • Dividing by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{1}{5^2} \times \frac{(-2)^3}{1} = \frac{(-2)^3}{5^2} \]

5. Calculating the values:

   • Now, we calculate \((-2)^3\) and \(5^2\): \[ (-2)^3 = -8 \] \[ 5^2 = 25 \]

6. Combining these values:

   • Now we can combine them into our fraction: \[ \frac{-8}{25} \]

Since we want the expression with only positive exponents and denominators, we can rewrite this as: \[ -\frac{8}{25} \]

However, if we're focusing on just the magnitude and ignoring the negative sign for the simplified fraction, the answer remains \(-\frac{8}{25}\). If you were looking for just the positive component in terms of its structure, the expression simplifies as shown.

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

Answer 10

idk if I really understood that bc I got it wrong but...
Apply the property of integer exponents to generate the equivalent expressions to (14^3 times 27^-7)^0 with only positive exponents then solve the expression. your answer will be a simplified fraction with no remaining exponent

Answer 11

I’m glad you’re continuing with your practice! Let's break down the expression \((14^3 \times 27^{-7})^0\) step by step using the properties of integer exponents.

1. Using the Zero Exponent Rule:

   • According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. Thus, we have: \[ (14^3 \times 27^{-7})^0 = 1 \]

2. Expressing with Positive Exponents:

   • Since \(1\) is already a simplified number and does not have any exponents to write as positive or negative, we don't need to do anything else to express it with only positive exponents.

3. Final Result:

   • Therefore, the entire expression simplifies to: \[ 1 \]

Conclusion: The final answer is: \[ 1 \] So, it seems there may have been a misunderstanding in the initial conditions, as this expression simplifies directly to the whole number \(1\). If you have any specific requirements for positive fractions, just let me know!