apply the properties of integer exponents to generate equivalent expressions to (4 ^ 3 * 7 ^ - 2)/(7 ^ 9 * 4 ^ - 13) 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 and generate equivalent expressions with only positive exponents, we can use the following rules:
1. Product Rule: a^m * a^n = a^(m + n)
2. Quotient Rule: a^m / a^n = a^(m - n)
3. Negative Exponent Rule: a^(-n) = 1 / a^n
Let's apply these rules step by step to simplify the given expression:
(4^3 * 7^(-2))/(7^9 * 4^(-13))
Using the product rule:
= (4^(3 - 13) * 7^(-2 + 9)) / 1 (since any number raised to the power of 0 is 1)
Simplifying further:
= (4^(-10) * 7^7) / 1
Applying the negative exponent rule:
= (1 / 4^10 * 7^7) / 1
Simplifying further:
= 1 / 4^10 * 7^7
Therefore, the simplified equivalent fraction is 1 / (4^10 * 7^7).
To apply the properties of integer exponents, let's start by simplifying the expression and converting all exponents to positive exponents.
Step 1: Rewrite the expression with positive exponents using the exponent properties:
(4^3 * 7^-2) / (7^9 * 4^-13)
Step 2: Apply the property of negative exponents; move each base with a negative exponent to the opposite position in the fraction and change the sign of the exponent:
(4^3 * 1/7^2) / (1/7^9 * 4^13)
Step 3: Multiply the numbers and raise the bases to their respective exponents:
(64 * 1/49) / (1/7^9 * 16^13)
Step 4: Simplify the fractions:
(64/49) / (1/7^9 * 16^13)
Step 5: Apply the property of multiplying fractions; multiply the numerator by the reciprocal of the denominator:
(64/49) * (7^9 * 16^13)
Step 6: Simplify further:
(64/49) * (7^9 * 2^4)^13
Step 7: Apply the property of multiplying powers with the same base; raise the base to the sum of the exponents:
(64/49) * (7^(9*13) * 2^(4*13))
Step 8: Simplify the exponents:
(64/49) * (7^117 * 2^52)
Now, the expression is simplified, but it is challenging to evaluate without specific values for the bases.