Apply the Properties of Integer Exponents to generate (2023)

1) Product Rule: (ab)^n = a^n * b^n
2) Power Rule: (a^n)^m = a^(n*m)
3) Negative Exponent Rule: a^(-n) = 1/a^n
4) Zero Exponent Rule: a^0 = 1

Let's apply these rules to the expression (143⋅27−7)034:

(143⋅27−7)034
= (143^0 * (27^-7))^34 (Using the zero exponent rule)

= 143^(0*34) * (27^-7*34) (By applying the power rule to the base)

= 143^0 * 27^(-7*34) (Simplifying the exponents)

= 1 * 27^(-7*34) (Using the zero exponent rule)

= 27^(-238) (Simplifying)

To solve the expression, we need to convert the exponent into a positive exponent. Using the negative exponent rule:

27^(-238) = 1/27^238

Therefore, the equivalent expression with positive exponents is 1/27^238.

The answer is 1/27^238, which is a simplified fraction with no remaining exponents.

To apply the properties of integer exponents and generate equivalent expressions with positive exponents, we can use the following rules:

1) Product Rule: (ab)^n = a^n * b^n
2) Power Rule: (a^n)^m = a^(n*m)
3) Negative Exponent Rule: a^(-n) = 1/a^n
4) Zero Exponent Rule: a^0 = 1

Let's apply these rules to the expression (143⋅27−7)034:

(143⋅27−7)034
= (143^0 * (27^(-7)))^34 (Using the zero exponent rule and negative exponent rule)

= 1^34 * (1/(27^7))^34

= 1 * (1^34/(27^7)^34) (Using the power rule)

= 1/(27^(7*34)) (Simplifying the exponents)

= 1/27^238

Therefore, the equivalent expression with positive exponents is 1/27^238.

The answer is 1/27^238, which is a simplified fraction with no remaining exponents.

1) Product Rule: (ab)^n = a^n * b^n
2) Power Rule: (a^n)^m = a^(n*m)
3) Negative Exponent Rule: a^(-n) = 1/a^n
4) Zero Exponent Rule: a^0 = 1

Let's apply these rules to the expression (43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13):

(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)
= (43^(-579) ⋅ (7^(-2))^(-579)) ⋅ (4^(-13))

= (43^(-579) ⋅ 7^(2*(-579))) ⋅ 4^(-13) (Applying the negative exponent rule)

= (43^(-579) ⋅ 7^(-1158)) ⋅ 4^(-13)

Now, let's simplify the exponents:

= (1/43^579) ⋅ (1/7^1158) ⋅ (1/4^13) (Applying the negative exponent rule)

= 1/(43^579 ⋅ 7^1158 ⋅ 4^13)

So, the equivalent expression with positive exponents is 1/(43^579 ⋅ 7^1158 ⋅ 4^13).

This expression cannot be simplified further.

Using the rule (a^m)^n = a^(m * n), we can simplify the exponents:

(43^(-579) ⋅ 7^(-1158)) ⋅ 4^(-13)
= (1 / 43^579) ⋅ (1 / 7^1158) ⋅ (1 / 4^13)

Using the rule (a / b)^n = a^n / b^n, we can combine the fractions:

= 1 / (43^579 ⋅ 7^1158 ⋅ 4^13)

Now, using the commutative property of multiplication, we can rearrange the terms:

= 1 / (43^579 ⋅ 4^13 ⋅ 7^1158)

Finally, using the associative property of multiplication, we can group the terms together:

= 1 / [(43 ⋅ 4)^13 ⋅ 7^1158]

= 1 / (172^13 ⋅ 7^1158)

Therefore, the simplified fraction with no remaining exponents is 1 / (172^13 ⋅ 7^1158).

Responses

1/5^4
Start Fraction 1 over 5 superscript 4 baseline end fraction

5^9/5^−13
Start Fraction 5 superscript 9 baseline over 5 superscript negative 13 baseline End Fraction

1/625
Start Fraction 1 over 625 end fraction

5^4

Let's use the negative exponent rule, which states that a^(-n) = 1/a^n.

59⋅5^−13 = 59/(5^13)

Therefore, the equivalent expression with positive exponents is 59/(5^13).

Out of the given responses, the equivalent expression that matches this form is:

5^9/5^−13 -> This is equivalent to 5^9 * 5^13 = 5^(9+13) = 5^22

Hence, the correct equivalent expression is 5^22.

Responses

1/5^4
Start Fraction 1 over 5 superscript 4 baseline end fraction

5^9/5^−13
Start Fraction 5 superscript 9 baseline over 5 superscript negative 13 baseline End Fraction

1/625
Start Fraction 1 over 625 end fraction

5^4

The rule we can apply here is the product rule, which states that a^m * a^n = a^(m+n).

Therefore, for 5^9⋅5^−13, we can combine the exponents:

5^9⋅5^−13 = 5^(9+(-13))

Simplifying the exponent, we get:

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

So, the equivalent expression with positive exponents is 5^(-4).

Out of the given responses, the equivalent expression that matches this form is:

1/5^4

Hence, the correct equivalent expression is 1/5^4.

Responses

only the Power Rule of Exponents
only the Power Rule of Exponents

only the Property of Negative Exponents
only the Property of Negative 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 Product Rule of Exponents

When dividing two expressions with the same base, we subtract the exponents. In this case, 3^5 divided by 3^−9 can be rewritten as 3^(5 - (−9)) = 3^(5 + 9) = 3^14.

Therefore, the correct response is:

the Property of Negative Exponents.

The Property of Negative Exponents and the Product Rule of Exponents were indeed used to generate the equivalent expression 3^14 from 3^5/3^−9.

Using the Property of Negative Exponents, we can rewrite 3^−9 as 1/3^9.
Then, using the Product Rule of Exponents, we can combine 3^5 and 1/3^9 by subtracting the exponents:
3^5 / 1/3^9 = 3^5 * 3^9 = 3^(5+9) = 3^14.

Therefore, the equivalent expression 3^14 was generated by using the Property of Negative Exponents and the Product Rule of Exponents.

Responses

7^2(−4)^3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction

(−4)^37^2
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction

4^3/7^2
Start Fraction 4 cubed over 7 squared end fraction

1/7^2 ⋅ (−4)^−3

The rule we can use here is the zero exponent rule, which states that a^0 = 1.

Also, using the negative exponent rule, we know that a^(-n) = 1/a^n.

Starting with the expression: 15^0 ⋅ 7^−2 / (−4)^−3,

15^0 = 1 (applying the zero exponent rule)

7^−2 = 1/7^2 (applying the negative exponent rule)

(−4)^−3 = 1/(−4)^3 = 1/(-4)^3

Therefore, the expression becomes:

1 ⋅ 1/7^2 / 1/(-4)^3

Simplifying further:

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

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

= 1/49 * (-4)^3

Out of the given responses, the equivalent expression that matches this form is:

4^3/7^2

Hence, the correct equivalent expression is 4^3/7^2.

Responses

60/4^−3
Start Fraction 60 over 4 negative cubed end fraction

1/64
Start Fraction 1 over 64 end fraction

1⋅4^−3
1 times 4 negative cubed

64

Starting with the expression: (15^-3 ⋅ 47)0^4^-3,

Let's simplify the exponents first:

15^-3 = 1/15^3

Now let's evaluate the expression further:

(1/15^3 ⋅ 47)0^(4^-3)

= (1/15^3 ⋅ 47)^0

Any expression raised to the power of 0 equals 1. Therefore:

(1/15^3 ⋅ 47)^0 = 1

Thus, the equivalent expression is 1.

Out of the given responses, the equivalent expression that matches this result is:

1

Hence, the correct equivalent expression is 1.

Responses

60/4^−3
Start Fraction 60 over 4 negative cubed end fraction

1/64
Start Fraction 1 over 64 end fraction

1⋅4^−3
1 times 4 negative cubed

64

Given the expression: (15^-3 ⋅ 4^7)^0/4^-3,

First, let's simplify the exponents within the parentheses:

15^-3 = 1/15^3

Next, we can rewrite the expression as (1/15^3 ⋅ 4^7)^0 / 4^-3.

Using the zero exponent rule, any expression raised to the power of 0 equals 1. Thus, (1/15^3 ⋅ 4^7)^0 simplifies to 1.

Now, we have 1 / 4^-3 left to evaluate.

Using the negative exponent rule, we can rewrite 4^-3 as 1/4^3.

Therefore, the expression becomes:

1 / 1/4^3 = 1 * 4^3 = 64.

Therefore, the equivalent expression is 64.

Out of the given responses, the equivalent expression that matches this result is:

64

Hence, the correct equivalent expression is 64.

Responses

2^3/9^2
Start Fraction 2 cubed over 9 squared end fraction

2^12⋅9^−12/2^9⋅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

2^3 ⋅9^2
2 cubed times 9 squared

2^−7⋅9^−1/2^9⋅9^−10

Given the expression: (2^-3 ⋅ 9^3)^-4/2^9 ⋅ 9^-10,

First, let's simplify the exponents within the parentheses:

2^-3 = 1/2^3 = 1/8

Next, we can rewrite the expression as ((1/8) ⋅ 9^3)^-4/2^9 ⋅ 9^-10.

Now, let's apply the power rule of exponents. Raising a power to another power requires multiplying the exponents. In this case, we have ((1/8)^1 ⋅ 9^3)^-4/2^9 ⋅ 9^-10.

Simplifying further:

((1/8)^1 ⋅ 9^3)^-4 = (1/8)^-4 ⋅ 9^-12 = 8^4 ⋅ 9^-12.

Now, let's simplify 2^9 ⋅ 9^-10:

2^9 ⋅ 9^-10 = 2^9 / 9^10.

Finally, we can combine the expressions:

(8^4 ⋅ 9^-12) / (2^9 / 9^10) = 8^4 ⋅ 9^-12 ⋅ 9^10 / 2^9.

Using the product rule of exponents, 9^-12 ⋅ 9^10 is equal to 9^(-12 + 10) = 9^-2 = 1/9^2.

Therefore, the expression becomes:

8^4 / (1/9^2 ⋅ 2^9) = 8^4 ⋅ 9^2 / 2^9.

Out of the given responses, the equivalent expression that matches this result is:

2^3/9^2.

Hence, the correct equivalent expression is 2^3/9^2.