Pls answer Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with positive exponents, we can use the following properties of integer exponents:
1) Product Rule: (a^m ⋅ b^n)^p = a^(m ⋅ p) ⋅ b^(n ⋅ p)
2) Quotient Rule: (a^m / b^n)^p = a^(m ⋅ p) / b^(n ⋅ p)
3) Power Rule: (a^m)^n = a^(m ⋅ n)
Using the product rule, we can rewrite the expression as:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Applying the power rule to (4^3 ⋅ 7^(-2))^(-5), we get:
4^(3 ⋅ -5) ⋅ 7^(-2 ⋅ -5) / 7^9 ⋅ 4^(-13)
Simplifying further, we have:
4^(-15) ⋅ 7^10 / 7^9 ⋅ 4^(-13)
Now, using the quotient rule, we can rewrite the expression as:
(4^(-15) / 4^(-13)) ⋅ (7^10 / 7^9)
Applying the power rule to 4^(-15) and 4^(-13), we get:
(4^(13-15) ⋅ 7^10 / 7^9)
Simplifying further:
4^(-2) ⋅ 7^10 / 7^9
Using the power rule to rewrite 4^(-2) as 1/4^2:
(1/4^2) ⋅ 7^10 / 7^9
Simplifying further:
1/16 ⋅ 7^10 / 7^9
Now, applying the power rule to 7^10 / 7^9, we get:
1/16 ⋅ 7^(10-9)
Simplifying further:
1/16 ⋅ 7^1
Finally, the expression simplifies to:
1/16 ⋅ 7
Which is equal to:
7/16
To generate equivalent expressions with positive exponents, we can apply the properties of integer exponents. Let's break down the given expression step by step:
Expression: (4^3 ⋅ 7^−2)^−5 / 7^9 ⋅ 4^−13
Step 1: Apply the Negative Exponent Rule
In this step, we will move any negative exponents to the denominator or vice versa by changing the sign of the exponent.
(4^3 / 7^2)^−5 / 7^9 ⋅ 4^−13
Step 2: Simplify the Exponent
We can simplify the exponent by multiplying it with the exponent outside the parentheses.
(4^(-15) / 7^(-10)) / 7^9 ⋅ 4^(-13)
Step 3: Apply the Quotient Rule for Exponents
To divide expressions with the same base, subtract the exponents.
4^((-15) - (-13)) / 7^((-10) - 9)
Simplifying further:
4^(-2) / 7^(-19)
Step 4: Apply the Negative Exponent Rule Again
In this step, we will move any negative exponents to the denominator or vice versa by changing the sign of the exponent.
7^19 / 4^2
Step 5: Simplify
Finally, we simplify the expression by evaluating the exponents:
7^19 = 1,470,084,984,576,699,049
4^2 = 16
So the simplified expression is:
1,470,084,984,576,699,049 / 16
To generate equivalent expressions with only positive exponents, we can make use of the properties of integer exponents. Let's break down the steps:
Step 1: Apply the properties of exponents within the parentheses.
(4^3 ⋅ 7^−2)^−5
Remember that when raising a power to another power, we multiply the exponents. So, here we get:
4^(3*(-5)) ⋅ 7^(-2*(-5))
Simplifying further:
4^(-15) ⋅ 7^(10)
Step 2: Apply the properties of exponents to the entire expression.
(4^(-15) ⋅ 7^(10)) / (7^9 ⋅ 4^(-13))
Here, dividing by a negative exponent is the same as multiplying by the positive exponent with the base flipped. Applying this, we get:
4^(-15) ⋅ 7^(10) / 7^9 ⋅ 4^(-13) / 1
Applying the division properties, we subtract the exponents:
4^(-15 - (-13)) ⋅ 7^(10 - 9)
Simplifying further:
4^(-2) ⋅ 7^1
Step 3: Solve the expression.
To simplify further, we evaluate the exponents:
4^(-2) = 1/4^2 = 1/16
7^1 = 7
So, the expression simplifies to:
(1/16) ⋅ 7
Multiply the numerator and denominator:
1/16 * 7/1 = 7/16
Therefore, the simplified fraction with no remaining exponents is 7/16.