Apply the Properties of Integer Exponents to generate equivalent expressions to (7to the power of −3)to the power of 7 times 7to the power of 20 divided by 6to the power of−2

Question

with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)

Answer 1

To simplify the expression \((7^{-3})^7 \times \frac{7^{20}}{6^{-2}}\) while applying the properties of integer exponents, let's break it down step by step.

1. Apply the Power of a Power Property: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

2. Combine the Exponents with \(7^{20}\): \[ 7^{-21} \times 7^{20} = 7^{-21 + 20} = 7^{-1} \]

3. Simplify the Division by a Negative Exponent: Dividing by \(6^{-2}\) is equivalent to multiplying by \(6^2\): \[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \times 6^2 = \frac{6^2}{7^1} \]

4. Calculate \(6^2\): \[ 6^2 = 36 \]

5. Combine the Expressions: So, we have: \[ \frac{6^2}{7^1} = \frac{36}{7} \]

The final expression, with only positive exponents, is \(\frac{36}{7}\).

Therefore, the answer is: \[ \frac{36}{7} \] (as an improper fraction).

Answer 2

Apply the product rule of integer exponents to find the numerical equivalent of 12 to the power of −5 times 12 to the power of 7
Show your work and explain how you used the product rule of integer exponents.(2 points)

Answer 3

To find the numerical equivalent of \( 12^{-5} \times 12^{7} \) using the product rule of integer exponents, follow these steps:

Step 1: Identify the Product Rule

The product rule of integer exponents states that when multiplying two expressions with the same base, you can add their exponents: \[ a^m \times a^n = a^{m+n} \] In this case, both expressions share the same base \(12\).

Step 2: Apply the Product Rule

Given: \[ 12^{-5} \times 12^{7} \] Applying the product rule, we get: \[ 12^{-5 + 7} \]

Step 3: Simplify the Exponent

Now calculate the exponent: \[ -5 + 7 = 2 \] So we have: \[ 12^{2} \]

Step 4: Calculate \(12^2\)

Now we compute \(12^2\): \[ 12^2 = 144 \]

Conclusion

Thus, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is: \[ \boxed{144} \]

In summary, we used the product rule of integer exponents to combine the powers of \(12\) by adding the exponents, yielding \(12^2\), which we then calculated to be \(144\).

Answer 4

The NASA Star Finder charts are constructed based on the visible night sky at 34˚ N latitude, about as far north of the equator as Los Angeles, California and Wichita Falls, Texas. Will the constellations on the Star Finder chart shift in the sky if you observe the night sky from further south?(1 point)
Responses

No, the motion of Earth cancels out with the motions of stars.
No, the motion of Earth cancels out with the motions of stars.

No, the location of the stars in the sky is fixed.
No, the location of the stars in the sky is fixed.

Yes, they will shift north.
Yes, they will shift north.

Yes, they will shift south.
Yes, they will shift south.