Apply the properties of integer exponents to generate equivalent expressions to 4⁰ • 5^-²/(-2)^-³ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
4⁰ • 5^-²/(-2)^-³ =
To apply the properties of integer exponents, we can use the following rules:
1. Any number raised to the power of 0 is always 1: a^0 = 1, where a is any real number.
2. A negative exponent can be converted to a positive exponent by taking the reciprocal of the base: a^(-n) = 1/a^n, where a is any nonzero real number and n is a positive integer.
Using these rules, we can rewrite the given expression:
4⁰ • 5^-²/(-2)^-³
= 1 • (1/5²)/(1/(-2)³) (Using rule 1 and rule 2)
= 1 • (1/25)/(1/(-8))
= 1 • (-8/25) (Simplifying the fractions)
Therefore, the equivalent expression in positive exponents is 1 • (-8/25) = -8/25.
Wrong
I apologize for the mistake. Let's try again:
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1. Any number raised to the power of 0 is always 1: a^0 = 1, where a is any nonzero real number.
2. A negative exponent can be converted to a positive exponent by taking the reciprocal of the base: a^(-n) = 1/a^n, where a is any nonzero real number and n is a positive integer.
3. When multiplying two numbers with the same base raised to different exponents, we can add the exponents: a^n * a^m = a^(n+m), where a is any nonzero real number and n, m are integers.
Let's apply these rules to the given expression:
4^0 • 5^(-2) / (-2)^(-3)
Using rule 1, we know that 4^0 is equal to 1:
1 • 5^(-2) / (-2)^(-3)
Now, let's apply rule 2 and convert the negative exponents to positive exponents:
1 • 1/5^2 / 1/(-2)^3
Simplifying the denominator, we have:
1 • 1/5^2 / 1/(-8)
Calculating the values of the exponent:
1 • 1/25 / 1/(-8)
Using rule 3, we know that dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the expression as:
1 • 1/25 * (-8/1)
Simplifying the fractions:
1/25 * (-8)
Multiplying the numerator:
-8/25
Therefore, the equivalent expression with only positive exponents is -8/25.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we will use the following rules:
1. Any non-zero number raised to the power of zero is always equal to 1: a^0 = 1.
2. To change a negative exponent to a positive exponent, we can reciprocate the base and change the sign of the exponent: a^-n = 1/a^n.
Using these rules, let's simplify the expression:
4⁰ • 5^-² / (-2)^-³
Step 1: Simplify 4⁰
Since any non-zero number raised to the power of zero is equal to 1, we can simplify 4⁰ to 1:
1 • 5^-² / (-2)^-³
Step 2: Change negative exponents to positive exponents
Using the rule mentioned earlier, we can change the negative exponents to positive exponents by reciprocating the bases and changing the sign of the exponents:
1 • (1/5²) / (1 / (-2)³)
Step 3: Simplify the exponents
Now, let's simplify the exponents:
1 • (1/25) / (1 / -8)
Step 4: Simplify the division
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
1 • (1/25) * (-8/1)
Step 5: Simplify the expression
Multiply the numerators and denominators:
-8/25
So, the simplified fraction form of the given expression is -8/25.
To generate equivalent expressions with only positive exponents, we can make use of the properties of integer exponents:
1. Any number raised to the power of 0 is equal to 1.
Therefore, 4⁰ = 1.
2. To change a negative exponent to a positive exponent, we can move the base to the denominator and change the sign of the exponent.
So, 5⁻² = 1/5² = 1/25.
3. To change the sign of a negative exponent, we can move the base to the denominator or numerator, depending on its current position.
So, (-2)⁻³ = 1/(-2)³ = 1/(-8) = -1/8.
Now let's substitute these values into the original expression:
4⁰ • 5⁻² / (-2)⁻³ = 1 • (1/25) / (-1/8)
Since division is equivalent to multiplying by the reciprocal, we can rewrite the expression as:
1 • (1/25) • (-8/1)
Simplifying further, we have:
(-8/25)
Therefore, the given expression simplifies to -8/25.