Question

^Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

4^0 ⋅ 5^−2/(−2)^−3=

Answer 1

[AI-generated answer — Bot GPT 3.5]

4^0 ⋅ 5^−2/(−2)^−3 = 1 ⋅ (1/5^2) / (1/(-2)^3)

Using the property (a/b)^n = (a^n)/(b^n),
= 1 ⋅ 1/(5^2) / 1/(-8)

Using the property a/b = a ⋅ 1/b,
= 1/(5^2) ⋅ (-8)

Simplifying further,
= 1/25 ⋅ (-8)

Multiplying,
= -8/25

Therefore, the expression simplifies to -8/25.

Answer 2

Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

(14^3⋅27^−7)^0/3^4=

Answer 3

[AI-generated answer — Bot GPT 3.5]

(14^3⋅27^−7)^0/3^4 = 1/3^4

Using the property (a^m)^n = a^(m*n),
= (14^3)^(0/1) ⋅ (27^−7)^(0/1) / (3^4)^(0/1)

Using the property a^0 = 1,
= (1) ⋅ (1) / (1)

Therefore, the expression simplifies to 1/1, which is equal to 1.

Answer 4

[AI-generated answer — Step-by-Step Bot]

To generate equivalent expressions with only positive exponents, we can make use of the following properties of integer exponents:

1. Any non-zero number raised to the power of zero is equal to 1.
(a^0 = 1, where a ≠ 0)

2. To convert a negative exponent to a positive exponent, we take the reciprocal of the base and make the exponent positive.
(a^(-n) = 1 / a^n, where a ≠ 0)

Now let's apply these properties to the given expression: 4^0 ⋅ 5^(-2) / (-2)^(-3)

Using the first property, 4^0 simplifies to 1.

Applying the second property to 5^(-2), we get 1 / 5^2.

Using the second property again for (-2)^(-3), we get 1 / (-2)^3.

Therefore, the expression becomes:
1 ⋅ 1 / 5^2 / 1 / (-2)^3

Simplifying further by dividing fractions:
1 / (1 / 5^2) ⋅ 1 / (1 / (-2)^3)

Multiplying fractions is done by multiplying the numerators and denominators separately:

1 / (1/25) ⋅ 1 / (1 / (-8))

Rewriting the divisions as multiplication:

1 ⋅ (25 / 1) ⋅ (-8 / 1)

Now, multiplying 1 by any number doesn't change its value, so we can ignore it:

(25 / 1) ⋅ (-8 / 1)

Multiplying the numerators and denominators:

(25 x -8) / (1 x 1)

Simplifying the numerator:

-200 / 1

Therefore, -200 is the simplified fraction with no remaining exponents.