Apply the Properties of Integer Exponents to generate equivalent expressions 3 ^ 7 * 3 ^ - 9 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, we can combine the two exponential terms using the rule that states when multiplying exponential terms with the same base, we add the exponents.
So, 3^7 * 3^-9 can be simplified as 3^(7 + (-9)).
3^(7 + (-9)) = 3^(-2)
Now, to convert this expression to positive exponents, we can use another property of exponents which states that any non-zero number raised to the power of -n is equal to 1 divided by that number raised to the power of n.
Therefore, 3^(-2) can be written as 1/3^2.
Simplifying further,
1/3^2 = 1/9
Thus, the simplified fraction with no remaining exponents for the expression 3^7 * 3^-9 is 1/9.
Apply the Properties of Integer Exponents to generate equivalent expressions to 222−4 with only positive exponents, then solve the expression. Provide the solution as a whole number.(1 point)
222−4=
To generate equivalent expressions with only positive exponents, we can use the properties of exponents. Let's start by applying the product of powers property:
a^m * a^n = a^(m + n)
In this case, we can apply the property to the expression 3^7 * 3^(-9):
3^7 * 3^(-9) = 3^(7 + (-9))
Now, simplify the exponent:
3^(7 + (-9)) = 3^(-2)
Next, we can use another property of exponents, the negative exponent property:
a^(-n) = 1 / a^n
Apply this property to 3^(-2):
3^(-2) = 1 / 3^2
Simplify the expression further:
1 / 3^2 = 1 / 9
Therefore, the simplified value of the expression 3^7 * 3^(-9) is 1/9.
To apply the properties of integer exponents to generate equivalent expressions, we use the following rules:
1. Product of Powers Rule: a^m * a^n = a^(m + n)
2. Quotient of Powers Rule: a^m / a^n = a^(m - n)
3. Power of a Power Rule: (a^m)^n = a^(m * n)
4. Negative Exponent Rule: a^(-n) = 1 / a^n
Let's apply these rules to the given expression, 3^7 * 3^(-9):
Step 1: Use the Product of Powers Rule to combine the exponents.
3^7 * 3^(-9) = 3^(7-9)
Step 2: Simplify the exponent.
3^(7-9) = 3^(-2)
Step 3: Apply the Negative Exponent Rule to change the exponent to a positive exponent.
3^(-2) = 1 / 3^2
Step 4: Simplify.
1 / 3^2 = 1 / 9
So, the simplified fraction form of the expression 3^7 * 3^(-9) with only positive exponents is 1/9.