Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 x 5^-2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To apply the properties of integer exponents to generate equivalent expressions, we can use the following rules:
1) a^0 = 1 (any nonzero number raised to the power of zero is equal to 1)
2) a^(-b) = 1/(a^b) (a negative exponent can be rewritten as the reciprocal of the positive exponent)
Now let's apply these rules to the given expression: 4^0 x 5^-2/(-2)^-3.
1) 4^0 is equal to 1, so we can rewrite the expression as 1 x 5^-2/(-2)^-3.
2) 5^-2 can be rewritten as 1/(5^2) and (-2)^-3 can be rewritten as 1/((-2)^3), so the expression becomes 1 x 1/(5^2) / 1/((-2)^3).
Now we can simplify the expression further. In order to divide fractions, we multiply by the reciprocal of the divisor. So the expression becomes:
1 x 1/(5^2) x ((-2)^3)/1.
Simplifying the fractions and exponents, we get:
1 x 1/(25) x (-8) = -8/25.
Therefore, the simplified fraction with no remaining exponents is -8/25.
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To apply the properties of integer exponents, we can rewrite the given expression as:
4^0 x 5^-2 / (-2)^-3
Recall that any non-zero number raised to the power of zero is equal to 1. Thus, 4^0 is 1.
1 x 5^-2 / (-2)^-3
Next, let's apply the property of negative exponents. Any term raised to a negative exponent can be rewritten as the reciprocal of that term raised to the positive exponent. So, 5^-2 can be written as 1/5^2, and (-2)^-3 can be written as 1/(-2)^3.
1 x (1/5^2) / (1/(-2)^3)
Simplifying further using the property of division, we can multiply the numerator by the reciprocal of the denominator:
(1 x (1/5^2)) x ((-2)^3 / 1)
This simplifies to:
(1/5^2) x (-2)^3
Now, let's simplify the exponents. 5^2 is equal to 25, and (-2)^3 is -8.
(1/25) x (-8)
Finally, we multiply the terms:
-8/25
Therefore, the simplified fraction is -8/25.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents for the expression 4^0 x 5^-2/(−2)^−3, we first need to understand the rules involved.
1. Any number raised to the power of 0 is equal to 1. Therefore, 4^0 = 1.
2. For negative exponents, we can use the rule that x^-n = 1/x^n. So, 5^-2 = 1/5^2 = 1/25.
3. To deal with the negative exponent in the denominator, we can apply the rule (1/x)^n = 1/x^n. So, (-2)^-3 = 1/(-2)^3 = 1/(-8) = -1/8.
Now, let's substitute these equivalent expressions back into the original expression:
1 x 1/25 / (-1/8)
To simplify this expression further, we need to remember that division by a fraction is the same as multiplying by its reciprocal. So, dividing by -1/8 is equivalent to multiplying by -8/1:
1 x 1/25 x (-8/1)
Multiplying the numerators and denominators separately, we get:
-8/25
Therefore, the simplified fraction for the expression 4^0 x 5^-2/(−2)^−3 is -8/25.