To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1, where a ≠ 0.
2. When dividing two numbers with the same base raised to different exponents, we subtract the exponents: a^m / a^n = a^(m - n), where a ≠ 0.
3. When raising a number with a negative exponent to the power of a positive exponent, we can move the base to the denominator and change the sign of the exponent: a^(-m) = 1/a^m, where a ≠ 0.
Let's apply these properties step by step:
First, we have 4^0 ⋅ 5^(-2)/(2^(-3)).
According to property 1, 4^0 = 1.
Next, we can apply property 3 to convert 5^(-2) to 1/5^2 = 1/25.
Similarly, we can apply property 3 to convert (2^(-3)) to 1/(2^3) = 1/8.
Now, our expression becomes 1 ⋅ 1/25 / 1/8.
Applying property 2, we subtract the exponents in the denominator when dividing: 1 ⋅ 1/25 * 8/1 = 8/25.
Finally, the simplified fraction with no remaining exponents is 8/25.
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..
5 answers
that was incorrect, try again
Apologies for the incorrect response. Let's try again.
To generate equivalent expressions with positive exponents, we can use the properties of integer exponents:
1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1, where a ≠ 0.
2. When dividing two numbers with the same base raised to different exponents, we subtract the exponents: a^m / a^n = a^(m - n), where a ≠ 0.
3. When raising a number with a negative exponent to the power of a positive exponent, we can move the base to the denominator and change the sign of the exponent: a^(-m) = 1/a^m, where a ≠ 0.
Let's apply these properties step by step:
The expression is 4^0 ⋅ 5^(-2)/(−2)^(-3).
According to property 1, 4^0 = 1.
Applying property 3, we can convert 5^(-2) to 1/5^2 = 1/25.
Similarly, we can convert (−2)^(-3) to 1/(−2)^3 = 1/(-8) = -1/8.
Now, the expression becomes 1 ⋅ 1/25 / (-1/8).
We can simplify this further. When dividing by a fraction, we multiply by its reciprocal. So, the expression becomes 1 ⋅ 1/25 * (-8/1) = -8/25.
Therefore, the simplified fraction with no remaining exponents is -8/25.
To generate equivalent expressions with positive exponents, we can use the properties of integer exponents:
1. Any non-zero number raised to the power of 0 is equal to 1: a^0 = 1, where a ≠ 0.
2. When dividing two numbers with the same base raised to different exponents, we subtract the exponents: a^m / a^n = a^(m - n), where a ≠ 0.
3. When raising a number with a negative exponent to the power of a positive exponent, we can move the base to the denominator and change the sign of the exponent: a^(-m) = 1/a^m, where a ≠ 0.
Let's apply these properties step by step:
The expression is 4^0 ⋅ 5^(-2)/(−2)^(-3).
According to property 1, 4^0 = 1.
Applying property 3, we can convert 5^(-2) to 1/5^2 = 1/25.
Similarly, we can convert (−2)^(-3) to 1/(−2)^3 = 1/(-8) = -1/8.
Now, the expression becomes 1 ⋅ 1/25 / (-1/8).
We can simplify this further. When dividing by a fraction, we multiply by its reciprocal. So, the expression becomes 1 ⋅ 1/25 * (-8/1) = -8/25.
Therefore, the simplified fraction with no remaining exponents is -8/25.
that was correct, thank you
You're welcome! I'm glad the second response was correct. If you have any more questions, feel free to ask.
1 answer