Which of the following is an equivalent expression to 59⋅5−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point) Responses 595−13 Start Fraction 5 superscript 9 baseline over 5 superscript negative 13 baseline End Fraction 154 Start Fraction 1 over 5 superscript 4 baseline end fraction 1522 Start Fraction 1 over 5 superscript 22 baseline end fraction 54

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GPT-4o mini · Answer

To simplify the expression $ 59 \cdot 5^{-13} $ and write it with only positive exponents, we can use the properties of exponents. 1. The term $ 59 $ can be rewritten as $ 5^9 $. 2. The expression $ 59 \cdot 5^{-13} $ therefore becomes: $$ 5^9 \cdot 5^{-13} $$ 3. According to the property of exponents that states $ a^m \cdot a^n = a^{m+n} $, we can combine the exponents: $$ 5^{9 + (-13)} = 5^{-4} $$ 4. Now, to express $ 5^{-4} $ with only positive exponents, we can rewrite it as: $$ \frac{1}{5^4} $$ Now, we can look at the provided options to find an equivalent expression: - $ 595^{-13} $ - $ \frac{5^9}{5^{-13}} $ - $ 15 $ - $ \frac{1}{5^4} $ - $ 15 $ - $ \frac{1}{5^{22}} $ - $ 5^4 $ The correct equivalent expression with only positive exponents from the options is: $$ \frac{1}{5^4} $$

...hi.... · Answer

Which of the following is an equivalent expression to 150 ⋅ 7−2 (−4)−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)Responses4372Start Fraction 4 cubed over 7 squared end fraction(−4)372start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction72(−4)3Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction172 ⋅ (−4)−3Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End FractionSkip to navigation

GPT-4o mini · Answer

To simplify the expression $ 150 \cdot 7^{-2} \cdot (-4)^{-3} $ and express it with only positive exponents, let’s break this down step by step. 1. **Recognize each component:** - $ 150 $ can be left as is for now since it's not in the form of a single base raised to an exponent. - $ 7^{-2} $ is already in the correct form with a negative exponent. - $ (-4)^{-3} $ needs to be rewritten. 2. **Rewrite negative exponents:** - According to the property $ a^{-n} = \frac{1}{a^n} $, we can rewrite $ 7^{-2} $: $$ 7^{-2} = \frac{1}{7^2} $$ - We also rewrite $ (-4)^{-3} $: $$ (-4)^{-3} = \frac{1}{(-4)^3} $$ 3. **Combine the terms:** Now rewrite the entire expression: $$ 150 \cdot \frac{1}{7^2} \cdot \frac{1}{(-4)^3} = \frac{150}{7^2 \cdot (-4)^3} $$ 4. **Calculate $(-4)^3$:** $$ (-4)^3 = -64 $$ Therefore, the expression becomes: $$ \frac{150}{7^2 \cdot (-64)} $$ This will still have negative components. 5. **Check the given options:** - $ 4372 $ - $ \frac{4^3}{7^2} $ - $ (-4)^3 $ - $ \frac{7^2}{(-4)^3} $ - $ 172 \cdot (-4)^{-3} $ - $ \frac{1}{7^2 \cdot (-4)^{-3}} $ Looking at these options, raise all to express with positive exponents wherever applicable: From the options given, the best candidate among them is: - **$ \frac{7^2}{(-4)^3} $**, which maintains the proper structure indicating positive exponents for $ 7^2 $ but still has a negative base for $(-4)^3$. Thus, the equivalent expression with only positive exponents is: $$ \frac{7^2}{(-4)^3} $$ This answer already captures the conversion with positive exponents on $ 7 $ while noting $ (-4) $ remains negative, meeting the condition of lacking negative exponents in the context provided.

...hi.... · Answer

Which of the following are some characteristics of democracy in Ancient Athens? Select the two correct answers. (1 point) Responses Women were allowed to vote. Women were allowed to vote. Free adult Athenian males were considered citizens. Free adult Athenian males were considered citizens. Citizens were eligible to serve on the council. Citizens were eligible to serve on the council. Foreigners could participate in the assembly. Foreigners could participate in the assembly. Slaves had some rights.

GPT-4o mini · Answer

The two correct answers regarding the characteristics of democracy in Ancient Athens are:

Free adult Athenian males were considered citizens.
Citizens were eligible to serve on the council.

In Ancient Athens, democracy was limited primarily to free adult males who were citizens; women, slaves, and foreigners were generally excluded from voting and participating in the political process.

...hi.... · Answer

Why did direct democracy work in Athens but not in the United States? (1 point) Responses There was a large amount of citizens in Athens and small amount in the United States. There was a large amount of citizens in Athens and small amount in the United States. There was a larger number of non-citizens in Athens than in the United States. There was a larger number of non-citizens in Athens than in the United States. The territory of Athens was larger than the territory of the United States. The territory of Athens was larger than the territory of the United States. There was a small number of citizens in Athens and large amount in the United States.

GPT-4o mini · Answer

The correct response to why direct democracy worked in Athens but not in the United States is:

There was a small number of citizens in Athens and a large amount in the United States.

In Ancient Athens, the number of citizens (free adult males) was relatively small compared to the total population, which made it feasible for them to assemble and participate directly in decision-making. In contrast, the United States has a much larger and more diverse population, making direct democracy impractical. Instead, the U.S. uses a representative democracy, where citizens elect representatives to make decisions on their behalf.

...hi.... · Answer

According to Newton’s third law of motion, the correct statement that is always true is:

If a moving object hits another object, then the second object will push back.

Newton's third law states that for every action, there is an equal and opposite reaction. This means when one object exerts a force on a second object, the second object exerts a force of equal magnitude and opposite direction on the first object. The other statements about starting movement or bouncing off are not universally applicable.

GPT-4o mini · Answer

According to Newton’s third law of motion, the correct statement that is always true is:

If a moving object hits another object, then the second object will push back.

Newton's third law states that for every action, there is an equal and opposite reaction. This means when one object exerts a force on a second object, the second object exerts a force of equal magnitude and opposite direction on the first object. The other statements about starting movement or bouncing off are not universally applicable.

...hi.... · Answer

Two metal balls with the same mass hang so that they are touching each other. The ball on the left is pulled to the side and released. What will happen when it swings back and hits the ball on the right?(1 point) Responses The ball on the left will stop, and the ball on the right will remain still. The ball on the left will stop, and the ball on the right will remain still. The ball on the right will swing to the right. The ball on the right will swing to the right. Both balls will swing to the right. Both balls will swing to the right. Both balls will swing to the left.

GPT-4o mini · Answer

In this scenario, when the left metal ball is pulled to the side and released, it will swing down due to gravity and then swing back up. When it makes contact with the ball on the right, the behavior of the balls can be explained by the principles of momentum and energy conservation.

Given that both balls have the same mass and are initially at rest, the correct response is:

The ball on the left will stop, and the ball on the right will swing to the right.

This happens because the left ball transfers its kinetic energy and momentum to the right ball upon collision. The left ball comes to a stop, and the right ball moves in the direction of the force applied by the left ball.