Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 =

Answer 1

[AI-generated answer — Bot GPT 3.5]

To apply the properties of integer exponents, we can use the following rules:

1. To simplify a negative exponent, we can take the reciprocal of the base and make the exponent positive.
2. When multiplying two expressions with the same base, we can add their exponents.
3. When dividing two expressions with the same base, we can subtract their exponents.

Using these rules, let's simplify the given expression step by step:

(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13

Step 1: Apply the negative exponent rule to (7^-2), making it (1/7^2):
(4^3 ⋅ (1/7^2))^-5/7^9 ⋅ 4^-13

Step 2: Multiply the exponents inside the parentheses (4^3 and 1/7^2):
4^(3 * -5)/7^(2 * -5) ⋅ 4^-13

Simplifying the exponents:
4^-15/7^-10 ⋅ 4^-13

Step 3: Apply the negative exponent rule to (4^-15), making it (1/4^15):
(1/4^15)/(7^-10) ⋅ 4^-13

Step 4: Divide the exponents (1/4^15 divided by 7^-10):
1/(4^15 * 7^-10) ⋅ 4^-13

Step 5: Apply the negative exponent rule to (7^-10), making it (1/7^10):
1/(4^15 * (1/7^10)) ⋅ 4^-13

Step 6: Multiply the expressions:
1/(4^15/7^10) ⋅ 4^-13

Step 7: Simplify 4^15/7^10 by subtracting the exponents:
1/(4^15/7^10) ⋅ 4^-13 = 1/((4^15)/(7^10)) ⋅ 4^-13

Step 8: Apply the negative exponent rule to (4^-13), making it (1/4^13):
1/((4^15)/(7^10)) ⋅ (1/4^13)

Step 9: Multiply the expressions:
1 * (1/((4^15)/(7^10) * 1/4^13)

Simplifying the multiplication:
1/(4^15/7^10 * 1/4^13)

Step 10: Multiply the fractions:
1/(4^15 * 7^10/4^13)

Step 11: Divide the exponents by subtracting them:
1/(4^(15-13) * 7^10)

Step 12: Simplify the exponents:
1/(4^2 * 7^10)

Step 13: Evaluate 4^2 and 7^10:
1/(16 * 282475249)

Step 14: Simplify the division:
1/4519603984

Therefore, (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = 1/4519603984.

Answer 2

(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13=

Answer 3

[AI-generated answer — Bot GPT 3.5]

To simplify the given expression, let's break it down step by step:

First, we can simplify the expression within the parentheses:

4^3 ⋅ 7^(-2) = 64 ⋅ 1/49 = 64/49

Now, let's substitute the simplified expression back into the original one:

(64/49)^(-5/7^9) ⋅ 4^(-13)

Next, we can apply the rule of negative exponents to the base 64/49:

(49/64)^(5/7^9) ⋅ 4^(-13)

Now, we can simplify the expression 4^(-13) using the rule of negative exponents:

(49/64)^(5/7^9) ⋅ 1/4^13

Lastly, we can combine the exponents of 49/64:

(49/64)^(5/7^9) ⋅ 1/2^26

So, the equivalent expression to (4^3 ⋅ 7^(-2))^(-5/7^9) ⋅ 4^(-13) is (49/64)^(5/7^9) ⋅ 1/2^26.

Answer 4

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

54
5 superscript 4 baseline

154
Start Fraction 1 over 5 superscript 4 baseline end fraction

1625
Start Fraction 1 over 625 end fraction

595−13

Answer 5

[AI-generated answer — Bot GPT 3.5]

To generate an equivalent expression with only positive exponents, we can apply the property of negative exponents, which states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent. However, it seems that the expression given, 59⋅5−13, already only has positive exponents. Therefore, none of the given responses is an equivalent expression generated by applying the Properties of Integer Exponents.

Answer 6

Which of the following is an equivalent expression to 5^9⋅5^−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)

Responses

a. 54
5 superscript 4 baseline

b. 154
Start Fraction 1 over 5 superscript 4 baseline end fraction

c. 1625
Start Fraction 1 over 625 end fraction

d. 595−13

Answer 7

[AI-generated answer — Bot GPT 3.5]

To find an equivalent expression with only positive exponents, we can use the property of product of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.

In the given expression, 5^9 * 5^(-13), we have the same base of 5. So, we can add the exponents:

5^(9 + (-13))

Simplifying the exponent gives us:

5^(-4)

Now, we can apply the property of negative exponents, which states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent:

1 / 5^4

Therefore, the equivalent expression to 5^9 * 5^(-13) with only positive exponents is:

1 / 5^4

Looking at the given responses:

a. 54
b. 154
c. 1625
d. 595^(-13)

None of the responses match the equivalent expression we found, which is 1 / 5^4.

Answer 8

wrong which one is the answer

Answer 9

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion. Let's go through the options again and find the correct equivalent expression.

The original expression is 5^9 * 5^(-13). To find the equivalent expression with only positive exponents, we can use the property of product of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.

In this case, we have:

5^9 * 5^(-13) = 5^(9 + (-13)) = 5^(-4)

Now, let's simplify the negative exponent. The property of negative exponents states that a negative exponent can be rewritten as the reciprocal with a positive exponent.

Therefore, 5^(-4) can be rewritten as:

1 / 5^4

Looking again at the given responses:

a. 54
b. 154
c. 1625
d. 595^(-13)

The correct answer is:

c. Start Fraction 1 over 625 end fraction

The equivalent expression to 5^9 * 5^(-13) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 1 over 625 end fraction.

Answer 10

Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point)

Responses

only the Property of Negative Exponents
only the Property of Negative Exponents

only the Product Rule of Exponents
only the Product Rule of Exponents

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

only the Power Rule of Exponents

Answer 11

[AI-generated answer — Bot GPT 3.5]

The equivalent expression was generated using the property of negative exponents.

In the original expression, 3^5/3^(-9), there is a negative exponent in the denominator. The property of negative exponents states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent.

Applying this property to the expression gives us:

3^5 * 3^9

Since two expressions with the same base are being multiplied, we can apply the product rule of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.

Therefore, the equivalent expression 3^14 was generated using only the property of negative exponents.

The correct response is:

only the Property of Negative Exponents

Answer 12

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

Responses

72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction

172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction

(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction

4372

Answer 13

[AI-generated answer — Bot GPT 3.5]

To find an equivalent expression with only positive exponents, we can apply the property of negative exponents, which states that a negative exponent can be rewritten as the reciprocal with a positive exponent. Additionally, any number raised to the power of 0 is equal to 1.

Let's simplify the given expression step by step:

1) 15^0 = 1 (any number raised to the power of 0 is equal to 1)

2) 7^(-2) = 1 / 7^2 (applying the property of negative exponents)

3) (-4)^(-3) = 1 / (-4)^3 (applying the property of negative exponents)

Now, we substitute these simplifications back into the original expression:

1 * (1 / 7^2) / (1 / (-4)^3)

Simplifying further:

1 * (1 / 7^2) * (-4)^3

Since raising -4 to the power of 3 is positive, we can leave it as is:

1 * (1 / 7^2) * (-4)^3

The result is:

(1 / 7^2) * (-4)^3

Comparing this result to the given responses:

1) 72(-4)^3
2) Start Fraction 7^2 / (-4)^3
3) (−4)^3 / 7^2
4) 4372

The correct answer is:

2) Start Fraction 7^2 / (-4)^3

The equivalent expression to 15^0 * 7^(-2) / (-4)^(-3) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 7^2 / (-4)^3.

Answer 14

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

Responses

64
64

604−3
Start Fraction 60 over 4 negative cubed end fraction

164
Start Fraction 1 over 64 end fraction

1⋅4−3

Answer 15

[AI-generated answer — Bot GPT 3.5]

To find the equivalent expression, let's simplify step by step while applying the properties of integer exponents:

1. Distribute the exponent of 0 to each term within the parentheses:

(15^-3 * 4^7)^0 = 1^0 = 1

2. Rewrite 4^-3 using the property of negative exponents:

4^-3 = 1/4^3

Now, substitute the simplified expressions back into the original expression:

1 / 4^3 = 1 / 64

Looking at the given options:

1) 64
2) Start Fraction 60 / 4^(-3)
3) Start Fraction 1 / 64
4) 1 * 4^(-3)

The correct answer is:

3) Start Fraction 1 / 64

This is the equivalent expression to (15^(-3) * 4^7)^0 / 4^(-3) that has been generated by applying the Properties of Integer Exponents.

Answer 16

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

Responses

212⋅9−1229⋅9−10
Start fraction 2 superscript 12 baseline times 9 superscript negative 12 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction

2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction

2392
Start Fraction 2 cubed over 9 squared end fraction

23 ⋅92

Answer 17

[AI-generated answer — Bot GPT 3.5]

To find the equivalent expression with only positive exponents, let's simplify the given expression step by step:

1. Apply the property of negative exponents to the base 2 and 9:

(2^-3 * 9^3)^-4 / 2^9 * 9^-10 = (1/2^3 * 9^3)^-4 / 2^9 * 1/9^10

2. Simplify the expressions within the parentheses using the property of negative exponents:

(1/8 * 9^3)^-4 / 2^9 * 1/9^10 = (9^3/8)^-4 / 2^9 * 1/9^10

3. Combine the exponents by applying the property of negative exponents and the product rule of exponents:

(9^(3*-4) / 8^-4) / 2^9 * 1/9^10 = (9^-12 / 8^-4) / 2^9 * 1/9^10

4. Simplify the negative exponents using the property of negative exponents:

(8^4/9^12) / 2^9 * 1/9^10

5. Combine the fractions by subtracting the exponents:

8^4 / 9^12 / (2^9 * 9^10)

6. Combine the bases with the same exponent:

8^4 / (2^9 * 9^22)

Therefore, the equivalent expression to (2^-3 * 9^3)^-4 / 2^9 * 9^-10 with only positive exponents is:

8^4 / (2^9 * 9^22)

Looking at the given responses:

1) Start fraction 2^12 * 9^-12 / 2^9 * 9^-10
2) Start fraction 2^-7 * 9^-1 / 2^9 * 9^-10
3) Start fraction 2^3 / 9^2
4) 2^3 * 9^2

The correct answer is:

3) Start fraction 2^3 / 9^2

This is the equivalent expression to (2^-3 * 9^3)^-4 / 2^9 * 9^-10 with only positive exponents, generated by applying the Properties of Integer Exponents.