Simplify the expression. Assume that all variables represent positive numbers. Write your answer without using negative exponents.
(343x^-6y/8y^-8x^6)^-2/3
(343x^-6y/8y^-8x^6)^-2/3
= (8y^-8x^6/343x^-6y)^(2/3)
= (8/343)^(2/3) (x^6/x^-6)^(2/3) (y^-8/y)^(2/3)
= (2/7)^2 (x^12)^(2/3) (y^-9)^(2/3)
= 4/49 x^8/y^6
or
4x^8 / 49y^6
To simplify the expression, we can start by distributing the exponent (-2/3) to each term inside the parentheses. Applying the power of a quotient rule, we can also multiply the exponents together.
First, let's simplify the numerator:
343x^(-6)y = 7^3(x^(-2))^3(y^1)
= 7^3 * x^(-2 * 3) * y^1
= 343 * x^(-6) * y
Now, let's simplify the denominator:
8y^(-8)x^6 = 2^3(y^(-2))^4(x^2)^3
= 2^3 * y^(-2 * 4) * x^(2 * 3)
= 8 * y^(-8) * x^6
Now we can rewrite the expression as:
(343x^(-6)y) / (8y^(-8)x^6) = (343 * x^(-6) * y) / (8 * y^(-8) * x^6)
Next, let's take the reciprocal of the denominator and change the sign of the exponent:
= (343 * x^(-6) * y) / (8 * (1/y^8) * (1/x^6))
= (343 * x^(-6) * y) / (8/y^8 * 1/x^6)
Now we can multiply the fractions by multiplying the numerators and denominators separately:
= (343 * x^(-6) * y) * (y^8/8) * (x^6/1)
Simplifying further by multiplying the coefficients and terms with the same base, we get:
= (343/8) * (y * y^8) * (x^(-6) * x^6)
= 42.875 * y^(1 + 8) * x^(6 + (-6))
= 42.875 * y^9 * x^0
Finally, any term raised to the power of zero is equal to 1, so x^0 = 1. Thus, the simplified expression is:
= 42.875 * y^9 * 1
= 42.875y^9
To simplify the given expression, we can follow these steps:
Step 1: Simplify the numerator:
a. Convert the negative exponents to positive exponents by moving the terms to the opposite side of the fraction. For x^-6, move it to the denominator and it becomes x^6. Similarly, for y^-8, move it to the denominator and it becomes y^8.
b. Simplify the numerator by multiplying the numerical coefficients and adding the exponents of x and y. In this case, (343/8) becomes 343/8, and (x^-6 * x^6) becomes x^0.
c. Simplify x^0, which is equal to 1. Therefore, the numerator simplifies to 343/8.
Step 2: Simplify the denominator:
a. Convert the negative exponents to positive exponents by moving the terms to the opposite side of the fraction. For y^-8, move it to the numerator, and it becomes y^8. Similarly, for x^6, move it to the numerator, and it becomes x^-6.
b. Simplify the denominator by multiplying the numerical coefficients and adding the exponents of x and y. In this case, (8/1) becomes 8, and (y^-8 * y^8) becomes y^0.
c. Simplify y^0, which is equal to 1. Therefore, the denominator simplifies to 8.
Step 3: Simplify the entire expression:
a. Substitute the simplified numerator (343/8) and denominator (8) back into the expression.
b. Apply the exponent (-2/3) to both the numerator and denominator. To do this, raise each to the power of -2/3.
c. To raise the numerator to the power of -2/3, take the cube root of the numerator, and then square it: (343/8)^(-2/3) = (cube root of 343/8)^2.
d. Simplify the numerator further. The cube root of 343 is 7, and the cube root of 8 is 2. Therefore, the numerator becomes (7/2)^2.
e. To raise the denominator to the power of -2/3, take the cube root of the denominator, and then square it: 8^(-2/3) = (cube root of 8)^2.
f. Simplify the denominator further. The cube root of 8 is 2. Therefore, the denominator becomes 2^2, which is equal to 4.
Finally, the simplified expression is (7/2)^2/4.