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

Answer 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.

Answer 2

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

Answer 3

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.