Apply the property of integer exponents to generate equivalent expression 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
wrong yet again!
(4^3 × 7^(-2))^(-5) / 7^9 × 4^(-13)
= (7^2/4^3)^5 / (7^9/4^13)
= 7^10/4^15 * 4^13/7^9
= 7/4^2
= 7/16
To generate an equivalent expression with only positive exponents, we can use 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 exponent property - When raising a power to a power, multiply the exponents.
= 4^(3 × -5/7^9) × 7^(-2 × -5/7^9) × 4^(-13 × -5/7^9)
Step 2: Simplify the exponents:
For 4:
3 × -5/7^9 = -15/7^9
For 7:
-2 × -5/7^9 = 10/7^9
For 4 again:
-13 × -5/7^9 = 65/7^9
Therefore, the equivalent expression with only positive exponents is:
= 4^(-15/7^9) × 7^(10/7^9) × 4^(65/7^9)
Step 3: Combine the expressions with the same bases:
= 4^(-15/7^9 + 65/7^9) × 7^(10/7^9)
Step 4: Simplify the exponents:
= 4^(50/7^9) × 7^(10/7^9)
Now, let's move to solve the expression by evaluating the powers:
The expression 4^(50/7^9) means raising 4 to the power of (50/7^9).
Similarly, the expression 7^(10/7^9) means raising 7 to the power of (10/7^9).
Unfortunately, this involves extremely large numbers and computation power to calculate precisely. As a result, it is not possible to provide the exact simplified fraction without any remaining exponents.
However, we can present the solution in a more manageable form:
= 4^(50/7^9) × 7^(10/7^9)
= Approximately 1 × 1 (since the exponents in both cases are too small)
= 1
Therefore, the simplified fraction without any remaining exponents is 1.
To generate an equivalent expression with only positive exponents, we can apply the property of integer exponents which states that for any nonzero real number a,
a^(-n) = 1/a^n
Let's apply this property to the given expression:
(4^3 × 7^(-2))^(-5) / 7^9 × 4^(-13)
= (4^3 × 7^(-2))^(-5) ÷ 7^9 × 4^(-13)
= (1 / (4^(-3) × 7^(2)))^(-5) ÷ 7^9 × 4^(-13)
Now, we can simplify the exponents using the properties of exponents:
= (1 / (1 / 4^3 × 1 / 7^(-2)))^(-5) ÷ 7^9 × 4^(-13)
= (4^3 × 7^2)^(-5) ÷ 7^9 × 4^(-13)
= (4^(3 × -5) × 7^(2 × -5)) ÷ 7^9 × 4^(-13)
= 4^(-15) × 7^(-10) ÷ 7^9 × 4^(-13)
Now, let's simplify the expression further by combining the similar bases:
= (4^(-15 - 13) × 7^(-10 - 9))
= 4^(-28) × 7^(-19)
To solve this expression, we need to convert it into a simplified fraction without remaining exponents:
= 1 / (4^28 × 7^19)
Therefore, the simplified expression is 1 / (4^28 × 7^19), which is a fraction with no remaining exponents.
Apologies for the previous incorrect response. Thank you for catching that mistake.
To correctly generate an equivalent expression with only positive exponents and then solve the expression, let's break it down step by step.
Given expression: (4^3 × 7^-2)^(-5) / 7^9 × 4^-13
Step 1: Apply the property of integer exponents to remove negative exponents:
(4^3 × 7^-2)^(-5) = 1 / (4^3 × 7^2)^5
Step 2: Simplify the numerator and denominator separately:
Numerator: 1 / (4^3 × 7^2)^5 = 1 / (64 × 49)^5 = 1 / 313,810,596,090,625
Denominator: 7^9 × 4^-13 = (7^9 / 4^13)
Step 3: Combine the numerator and denominator:
(1 / 313,810,596,090,625) ÷ (7^9 / 4^13)
Step 4: Convert the division to multiplication by taking the reciprocal of the denominator:
(1 / 313,810,596,090,625) × (4^13 / 7^9)
Step 5: Simplify the exponents:
(1 / 313,810,596,090,625) × (4^13 / 7^9) = (1 / 313,810,596,090,625) × (2^26 / 7^9)
Step 6: Combine the numerator and denominator:
(1 × 2^26) / (313,810,596,090,625 × 7^9)
Step 7: Simplify further:
2^26 / (313,810,596,090,625 × 7^9) = 2^26 / 3,130,933,802,752,109,375
Step 8: Express the answer as a simplified fraction:
2^26 / 3,130,933,802,752,109,375 = 67,108,864 / 3,130,933,802,752,109,375
Therefore, the simplified fraction form of the expression is 67,108,864 / 3,130,933,802,752,109,375.