Apply the properties of integer exponents to generate equivalent expressions to (4^3 x 7^-2)^-5/7^9 x 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 only positive exponents, we can utilize the following properties of integer exponents:
1. Product of powers property: (a^m * a^n) = a^(m+n)
2. Quotient of powers property: (a^m / a^n) = a^(m-n)
3. Power of a power property: (a^m)^n = a^(m*n)
4. Negative exponent property: a^(-n) = 1 / a^n
5. Zero exponent property: a^0 = 1
Let's apply these properties to the given expression step by step:
(4^3 * 7^-2)^(-5) / 7^9 * 4^13
Step 1: Apply power of a power property
((4^3)^(-5) * (7^-2)^(-5)) / 7^9 * 4^13
Step 2: Simplify the exponents inside parentheses
(4^(-15) * 7^10) / 7^9 * 4^13
Step 3: Apply the quotient of powers property
4^(-15) * 7^(10-9) * 4^13
Step 4: Simplify the exponents
4^(-15) * 7^1 * 4^13
Step 5: Apply the product of powers property
4^(-15 + 13) * 7^1
Step 6: Simplify the exponent
4^(-2) * 7
Step 7: Apply the negative exponent property
1 / 4^2 * 7
Step 8: Calculate the values
1/16 * 7 = 7/16
Therefore, the final result is 7/16.
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To begin, let's simplify the given expression step by step using the properties of integer exponents:
Step 1: Apply the properties of exponents within parentheses.
(4^3 x 7^-2)^(-5/7^9 x 4^13)
Step 2: Simplify the exponents inside the parentheses:
(64 x 7^(-2))^(-5/7^9 x 4^13)
Step 3: Use the property of negative exponents to rewrite 7^(-2) as 1/7^2:
(64 x 1/7^2)^(-5/7^9 x 4^13)
Step 4: Simplify the expression inside the parentheses:
(64/49)^(-5/7^9 x 4^13)
Step 5: Apply the property of exponents when raising a power to another power:
(64/49)^(-5/7^9 x 4^13)
Step 6: Rewrite 4^13 as (2^2)^13 = 2^(2*13) = 2^26:
(64/49)^(-5/7^9 x 2^26)
Step 7: Apply the property of exponents on 2^26:
(64/49)^(-5/134217728 x 2^26)
Step 8: Convert 7^9 x 2^26 to a single exponent by multiplying:
(64/49)^(-5/134217728 x 2^26) = (64/49)^(-5/134217728 * 2^26)
Step 9: Simplify the exponent -5/134217728 * 2^26 = -5/134217728 * 2^26 = -5/134217728 * (2^13)^2 = -5/134217728 * 2^26 = -5/134217728 * 2^(13*2) = -5/134217728 * 2^26 = -5/134217728 * 64 = -320/134217728
Step 10: Substitute the simplified exponent back into the expression:
(64/49)^(-320/134217728)
Step 11: Apply the property of negative exponents to reverse the fraction:
(49/64)^(320/134217728)
Step 12: Simplify the exponent 320/134217728 = 5/209715456:
(49/64)^(5/209715456)
Therefore, the simplified expression is (49/64)^(5/209715456).
To generate equivalent expressions with only positive exponents, we need to apply the properties of integer exponents. Let's start with the given expression:
(4^3 x 7^-2)^-5/7^9 x 4^13
To simplify this expression, we'll work step by step:
Step 1: Applying the properties of negative exponents
Inside the parentheses, we have (4^3 x 7^-2). To apply the property of negative exponents, we move the base with a negative exponent from numerator to denominator, and change the sign of the exponent:
(4^3 x 1/7^2)^-5/7^9 x 4^13
Simplifying further:
(64 x 1/49)^-5/7^9 x 4^13
Step 2: Simplifying the expression inside the parentheses
Multiplying 64 by 1/49 gives us:
(64/49)^-5/7^9 x 4^13
Step 3: Applying the property of negative exponents again
Now, let's address the term (64/49)^-5. To apply the property of negative exponents, we move the base with a negative exponent from denominator to numerator, and change the sign of the exponent:
(49/64)^5/7^9 x 4^13
Step 4: Combining like terms
Now we have (49/64)^5/7^9 multiplied by 4^13. To combine them with only positive exponents, we'll multiply the exponents:
(49/64)^5/7^9 x 4^13 = (49^5 x 4^13)/(64^5 x 7^9)
Step 5: Simplifying the expression
Now we can evaluate the expression:
(49^5 x 4^13)/(64^5 x 7^9) =
5764801 x 67108864 / (1073741824 x 40353607) =
2676216323072 / 4323455642270007
Therefore, the simplified fraction of the given expression is 2676216323072/4323455642270007.