Which of the following is an equivalent expression to 5^9 times 5^-13 with only positive exponents that has been generated by applying the Properties of Integer Exponent?

Question

5^9/5^-13
5^4
1/5^22
1/5^4

Answer 1

To simplify the expression \( 5^9 \times 5^{-13} \), we can use the property of exponents that states \( a^m \times a^n = a^{m+n} \).

Here, we have:

\[ 5^9 \times 5^{-13} = 5^{9 + (-13)} = 5^{9 - 13} = 5^{-4}. \]

Now, to express this with only positive exponents, we can use the property that states \( a^{-n} = \frac{1}{a^n} \).

Thus, we can rewrite \( 5^{-4} \) as:

\[ 5^{-4} = \frac{1}{5^4}. \]

Now, let's evaluate the options provided:

1. \( 5^9/5^{-13} \): This can be simplified to \( 5^{9 - (-13)} = 5^{9 + 13} = 5^{22} \) but is not equivalent to \( 5^{-4} \).

2. \( 5^4 \): This is not equivalent to \( 5^{-4} \).

3. \( 1/5^{22} \): This is also not equivalent to \( 5^{-4} \).

4. \( 1/5^4 \): This matches \( 5^{-4} \).

Therefore, the equivalent expression to \( 5^9 \times 5^{-13} \) with only positive exponents is

\[ \boxed{1/5^4}. \]

Answer 2

Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9?
1. only the Product Rule of Exponents
2. only the Property of Negative Exponents
3. the Property of Negative Exponents and the Product Rule of Exponents
4. only the Power Rule of Exponents

Answer 3

To determine which properties of exponents were used to generate the expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), let's analyze the simplification step by step.

Starting with the expression:

\[ \frac{3^5}{3^{-9}} \]

We can apply the Quotient Rule of Exponents, which states that:

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

Using this rule, we subtract the exponents:

\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

Additionally, the step that involves moving \( 3^{-9} \) from the denominator to the numerator results in a positive exponent, which makes use of the Property of Negative Exponents that states:

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

So, in this case, we applied both the Property of Negative Exponents and the Quotient Rule of Exponents.

Therefore, the correct answer is:

3. the Property of Negative Exponents and the Product Rule of Exponents.

However, please note that the "Product Rule of Exponents" does not apply here directly, but rather we used the Quotient Rule specifically, so based on the listed options, it might be a potential oversight in terminology. The most accurate choice would mention the Quotient Rule instead.

Answer 4

Which of the following is an equivalent expression to 15^0 times 7^-2/(-4)^-3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
7^2/(-4)^-3
1/7^2 times (-4)^-3
4^3/7^2
(-4)^3/7^2

Answer 5

Let's simplify the expression \( 15^0 \times \frac{7^{-2}}{(-4)^{-3}} \) step by step, applying the properties of integer exponents.

1. Evaluate \( 15^0 \): According to the property of exponents, any non-zero number raised to the power of 0 is 1: \[ 15^0 = 1. \]

2. Rewrite the initial expression: The expression simplifies to: \[ 1 \times \frac{7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}}. \]

3. Apply the Quotient Rule of Exponents: We can rewrite \( \frac{a^m}{a^n} \) as \( a^{m-n} \). Here, we can convert the negative exponents: \[ \frac{7^{-2}}{(-4)^{-3}} = 7^{-2} \times (-4)^{3} = \frac{(-4)^{3}}{7^{2}}. \]

So we have transformed the expression to: \[ \frac{(-4)^3}{7^2}. \]

Now let's evaluate the options provided:

1. \( \frac{7^2}{(-4)^{-3}} \): This does not match our simplification.
2. \( \frac{1}{7^2} \times (-4)^{-3} \): This is an incorrect form because it does not combine the expression.
3. \( \frac{4^3}{7^2} \): This is not correct, since we have \( (-4)^3 \) not \( 4^3 \).
4. \( \frac{(-4)^3}{7^2} \): This matches our result from simplification.

Thus, the correct equivalent expression to \( 15^0 \times \frac{7^{-2}}{(-4)^{-3}} \) with only positive exponents is:

\[ \boxed{\frac{(-4)^3}{7^2}}. \]

Answer 6

Which of the following is the equivalent expression to (15^-3 times 4^7)^0/4^-3 that has been generated by applying the Properties of Integer Exponents?
1 times 4^-3
64
60/4^-3
1/64

Answer 7

To simplify the expression \(\frac{(15^{-3} \times 4^{7})^{0}}{4^{-3}}\), we will apply the properties of exponents.

1. Evaluate \((15^{-3} \times 4^{7})^{0}\): According to the property of exponents, any non-zero number (or expression) raised to the power of 0 is equal to 1. Therefore: \[ (15^{-3} \times 4^{7})^{0} = 1. \]

2. Rewrite the expression: Now we can substitute this back into the original expression: \[ \frac{1}{4^{-3}}. \]

3. Apply the property of negative exponents: The property of negative exponents states that \( a^{-n} = \frac{1}{a^n} \). Therefore: \[ \frac{1}{4^{-3}} = 4^{3}. \]

4. Calculate \(4^3\): Now calculating \(4^3\): \[ 4^{3} = 64. \]

Now, let’s evaluate the provided options:

1. \(1 \times 4^{-3}\): This simplifies to \(\frac{1}{4^3}\), which is not the result we found.
2. \(64\): This is exactly what we found, as \(4^3 = 64\).
3. \(\frac{60}{4^{-3}}\): This does not simplify to our result.
4. \(\frac{1}{64}\): This is also not correct, since we found \(64\).

Therefore, the correct answer is:

\[ \boxed{64}. \]

Answer 8

Which of the following is an equivalent expression to (2^-3 times 9^3)^-4/2^9 times 9^-10 with only positive exponents, generated by applying the Properties of Integer Exponents? 2^-7 times 9^-1/2^9 times 9^-10
2^3 times 9^2
2^3/9^2
2^12 times 9^-12/2^9 times 9^-10

Answer 9

To simplify the expression \(\frac{(2^{-3} \times 9^{3})^{-4}}{2^{9} \times 9^{-10}}\), we will follow the properties of exponents step by step.

1. Simplify \((2^{-3} \times 9^{3})^{-4}\): According to the Power Rule of Exponents, \((a^m \times b^n)^p = a^{mp} \times b^{np}\). Thus, we have: \[ (2^{-3} \times 9^{3})^{-4} = 2^{-3 \cdot -4} \times 9^{3 \cdot -4} = 2^{12} \times 9^{-12}. \]

2. Rewrite the entire expression: Now substituting this back into the original expression gives us: \[ \frac{2^{12} \times 9^{-12}}{2^{9} \times 9^{-10}}. \]

3. Apply the Quotient Rule of Exponents: Using the Quotient Rule of Exponents, \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify each factor: \[ = 2^{12-9} \times 9^{-12 - (-10)} = 2^{3} \times 9^{-12 + 10} = 2^{3} \times 9^{-2}. \]

4. Express with only positive exponents: The expression \(9^{-2}\) can be rewritten using the property of negative exponents: \[ 9^{-2} = \frac{1}{9^{2}}. \] So, the overall expression becomes: \[ 2^{3} \times \frac{1}{9^{2}} = \frac{2^{3}}{9^{2}}. \]

Now, let's evaluate the options provided:

1. \(2^{-7} \times 9^{-1}/(2^{9} \times 9^{-10})\): This expression does not reduce cleanly to what we obtained.
2. \(2^3 \times 9^2\): This is not correct; we have a \(9^{-2}\) in the expression.
3. \(\frac{2^3}{9^2}\): This matches our simplified result.
4. \(2^{12} \times 9^{-12}/(2^{9} \times 9^{-10})\): While this form may appear similar, it has not been further simplified or expressed as a positive exponent only.

The correct equivalent expression is:

\[ \boxed{\frac{2^3}{9^2}}. \]