Apply the Properties of Integer Exponents to generate an equivalent expression to 8^0 4^3
/ 5^-2with only positive exponents, then solve the expression. (1 point)
To generate an equivalent expression with only positive exponents, we can use the property that any number raised to the power of 0 is equal to 1. So, we have:
8^0 = 1
4^3 = 4 * 4 * 4 = 64
5^-2 = 1 / 5^2 = 1 / (5 * 5) = 1 / 25
Therefore, the equivalent expression is:
1 * 64 / (1 / 25) = 64 * 25 = 1600.
So the solution is 1600.
To generate an equivalent expression with only positive exponents, we can use the following properties of integer exponents:
1. Any number raised to the power of 0 is equal to 1.
2. To change the sign of a negative exponent, we can move the base to the opposite position and change the sign of the exponent.
Let's apply these properties to the given expression step by step:
1. Since 8^0 is equal to 1, we replace 8^0 with 1:
1 * 4^3 / 5^-2
2. To change the negative exponent 5^-2, we can move the 5 to the denominator and change the sign of the exponent:
1 * 4^3 * 5^2
Now, we have an equivalent expression with only positive exponents. To solve this expression, we simply perform the operations from left to right:
1. Evaluate the exponent of 4^3:
1 * 64 * 5^2
2. Evaluate the exponent of 5^2:
1 * 64 * 25
3. Perform multiplication:
1 * 1600
The final simplified expression is 1600.
To apply the properties of integer exponents, we can use the following rules:
1) Any number raised to the power of zero is equal to 1.
2) When multiplying two numbers with the same base but different exponents, you can add the exponents.
3) When dividing two numbers with the same base but different exponents, you can subtract the exponents.
4) A negative exponent can be changed to a positive exponent by taking the reciprocal of the base.
Given the expression:
8^0 * 4^3 / 5^-2
Let's apply the properties of integer exponents step by step:
Step 1: Simplify 8^0
By using rule 1, any number raised to the power of zero is equal to 1:
8^0 = 1
Step 2: Simplify 5^-2
By using rule 4, we can change the negative exponent to a positive exponent by taking the reciprocal of the base:
5^-2 = 1 / 5^2
Step 3: Simplify 4^3
No further simplification can be done here.
Now let's rewrite the expression using the simplified forms:
1 * 4^3 / (1 / 5^2)
Step 4: Multiply 1 and 4^3
1 * 4^3 = 4^3
Step 5: Divide by (1 / 5^2)
Dividing by a fraction is the same as multiplying by its reciprocal:
4^3 * (5^2 / 1)
Step 6: Simplify 4^3 and 5^2
4^3 = 64
5^2 = 25
Now we can substitute the values:
64 * (25 / 1)
Step 7: Multiply 64 and (25 / 1)
Multiplying two numbers is simply multiplying their values:
64 * 25 = 1600
Therefore, the equivalent expression with only positive exponents is 1600.