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.(1 point)
To apply the properties of integer exponents, we will use the following rules:
1. Product Rule: \(a^m \cdot a^n = a^{m+n}\)
2. Quotient Rule: \(a^m \div a^n = a^{m-n}\)
3. Power Rule: \((a^m)^n = a^{mn}\)
4. Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
Using these rules, we can rewrite the expression with only positive exponents:
\((4^3 ⋅ 7^{-2})^{-5} ⋅ 4^{-13} = \frac{1}{4^{3 \cdot 5}} \cdot \frac{1}{7^{-2 \cdot 5}} \cdot \frac{1}{4^{13}}\)
Simplifying the exponents:
\(\frac{1}{4^{15}} \cdot \frac{1}{7^{-10}} \cdot \frac{1}{4^{13}} = \frac{1}{4^{15} \cdot 7^{-10}} \cdot 4^{-13}\)
Now, using the product rule:
\(\frac{1}{4^{15} \cdot 7^{-10}} \cdot 4^{-13} = \frac{1}{4^{15 + (-13)} \cdot 7^{-10}}\)
Simplifying the exponents:
\(\frac{1}{4^{2} \cdot 7^{-10}} = \frac{1}{16 \cdot 7^{-10}}\)
Now, using the quotient rule:
\(\frac{1}{16 \cdot 7^{-10}} = \frac{1}{16} \cdot \frac{1}{7^{-10}}\)
Using the negative exponent rule:
\(\frac{1}{16} \cdot 7^{10}\)
Finally, simplifying the expression:
\(\frac{1}{16} \cdot 7^{10} = \frac{7^{10}}{16}\)
Therefore, the simplified fraction with no remaining exponents is \(\frac{7^{10}}{16}\).
wrong
I apologize for the mistake. Let's try to solve the expression correctly.
The expression is: \((4^3 \cdot 7^{-2})^{-5} \div (7^9 \cdot 4^{-13})\).
To simplify the expression, we can apply the properties of integer exponents:
Using the properties of exponentiation with negative exponents:
\(4^3 \cdot 7^{-2} = \frac{4^3}{7^2}\)
Using the properties of exponentiation with negative exponents:
\(7^9 \cdot 4^{-13} = \frac{7^9}{4^{13}}\)
Substituting these simplifications back into the original expression:
\((\frac{4^3}{7^2})^{-5} \div (\frac{7^9}{4^{13}})\)
Using the power rule for negative exponents:
\(\frac{1}{(\frac{4^3}{7^2})^5} \div (\frac{7^9}{4^{13}})\)
Using the power rule:
\(\frac{1}{(\frac{4^3}{7^2})^5} \div \frac{7^9}{4^{13}} = \frac{1}{(\frac{4^3}{7^2})^5} \cdot \frac{4^{13}}{7^9}\)
Simplifying the expression inside the parentheses:
\(\frac{1}{(\frac{64}{49})^5} \cdot \frac{4^{13}}{7^9}\)
Using the power rule:
\(\frac{1}{\frac{64^5}{49^5}} \cdot \frac{4^{13}}{7^9} = \frac{49^5}{64^5} \cdot \frac{4^{13}}{7^9}\)
Simplifying the fraction:
\(\frac{49^5 \cdot 4^{13}}{64^5 \cdot 7^9}\)
Using the power rule:
\(\frac{49^5 \cdot 2^{26}}{2^{30} \cdot 7^9}\)
Dividing the exponents with the same base:
\(\frac{49^5 \cdot 2^{26}}{2^{26} \cdot 2^{4} \cdot 7^9}\)
Simplifying the exponents:
\(\frac{49^5 \cdot 2^{26}}{2^{30} \cdot 2^{4} \cdot 7^9} = \frac{49^5 \cdot 2^{26}}{2^{34} \cdot 7^9}\)
Using the power rule:
\(\frac{49^5 \cdot 2^{26}}{(2 \cdot 7)^{9} \cdot 2^{34}}\)
Simplifying:
\(\frac{49^5 \cdot 2^{26}}{14^9 \cdot 2^{34}}\)
Using the power rule:
\(\frac{49^5}{14^9} \cdot \frac{2^{26}}{2^{34}}\)
Simplifying the exponents:
\(\frac{49^5}{14^9} \cdot 2^{-8}\)
Using the power rule:
\(\frac{49^5}{14^9} \cdot \frac{1}{2^8}\)
Simplifying:
\(\frac{49^5}{14^9 \cdot 2^8}\)
This is the simplified expression as a fraction with no remaining exponents.
To apply the properties of integer exponents, let's start by generating equivalent expressions with only positive exponents.
1. (4^3 ⋅ 7^−2)−5^/7^9 ⋅ 4^−13
First, let's simplify the exponents within the parentheses:
= (64 ⋅ 1/49)−5^/7^9 ⋅ 1/4^13
Next, let's simplify the exponents outside the parentheses:
= 64/49−5^/7^9 ⋅ 1/4^13
To simplify the expression further and remove negative exponents, we can use the following properties:
1. a^n / a^m = a^(n-m)
2. 1 / a^n = a^(-n)
Now let's simplify the expression using these properties:
= 64/49 ⋅ 7^(-5)/7^9 ⋅ 4^(-13)
= 64/49 ⋅ 7^(-5-9) ⋅ 4^(-13)
= 64/49 ⋅ 7^(-14) ⋅ 4^(-13)
To solve the expression, we need to simplify further:
= 64/49 ⋅ (1/7^14) ⋅ (1/4^13)
= 64/49 ⋅ (1/7^14) ⋅ (1/2^26)
To simplify, let's combine and simplify the fractions:
= (64/49) / (7^14 ⋅ 2^26)
Finally, let's rewrite the expression as a simplified fraction with no remaining exponents:
= 64 / (49 ⋅ 7^14 ⋅ 2^26)
And this is the simplified fraction with no remaining exponents.