Simplify the expression. Write your answer using only positive exponents.

Explain as well.

1. (8x)^3*8^-4
2. 2x^-5*y^-3
3. (4x^4y^-3)^-2
4. 1/(3x)^-2

1. (8x)^3*8^-4 = 8^3*x^3/8^4 = x^3/8

4. 1/(3x)^-2 = (3x)^2 = 9x^2

Now you try the others. Get rid of negative exponents by moving those terms from numerator to denominator, or vice versa.

I see that Reiny already answered all the questions for you, when you posted earlier.

1. To simplify the expression (8x)^3 * 8^-4, we first raise 8 to the power of 3, which gives us 8^3 = 8 * 8 * 8 = 512. We then multiply that result by 8 raised to the power of -4, which is 8^-4 = 1/8^4 = 1/4096.

So, (8x)^3 * 8^-4 simplifies to 512x^3/4096, which can be further simplified to x^3/8.

2. For the expression 2x^-5 * y^-3, we can rewrite it as 2/x^5 * 1/y^3.

This can be simplified by bringing the negative exponents to the denominator and changing the sign of the exponent:

2/(x^5 * y^3).

So, the simplified expression is 2/(x^5 * y^3).

3. For the expression (4x^4 * y^-3)^-2, we can first multiply the exponents inside the parentheses:

(4^(-2) * x^(4 * -2) * y^(-3 * -2)).

Simplifying this, we get:

(1/4^2 * x^-8 * y^6).

Further simplifying, we know that 4^2 is 16:

(1/16 * 1/x^8 * y^6).

So, the simplified expression is 1/16x^8y^6.

4. The expression 1/(3x)^-2 can be written as 1/(1/(3x)^2) by bringing the negative exponent inside the parentheses:

1/1/(3x)^2.

To simplify further, we can rewrite it as the reciprocal of (3x)^2:

1/(3x)^2 = (1/(3x))^2.

So, the simplified expression is (1/(3x))^2.

1. To simplify the expression (8x)^3 * 8^-4, we can use the properties of exponents. First, we can expand the expression (8x)^3 as (8^3)(x^3) since raising a power to a power means multiplying the exponents. So, now we have (8^3)(x^3) * 8^-4.

Next, we can simplify (8^3) as 512 (since 8^3 = 8 * 8 * 8 = 512). Our expression becomes 512(x^3) * 8^-4.

Using the property that a negative exponent represents taking the reciprocal, we can rewrite 8^-4 as 1/(8^4). Our expression now becomes 512(x^3) * 1/(8^4).

Simplifying further, we can evaluate 8^4 as 4096 (since 8^4 = 8 * 8 * 8 * 8 = 4096). Our expression is now 512(x^3)*(1/4096).

Finally, we can simplify by dividing 512 by 4096, which gives us 1/8. So the simplified expression becomes (1/8)(x^3) or (x^3)/8.

2. To simplify the expression 2x^-5 * y^-3, we can use the property that multiplying two numbers with the same base but different exponents is equivalent to adding the exponents. So, the expression becomes 2 * (x^-5) * (y^-3).

Now, using the rule that a negative exponent represents taking the reciprocal, x^-5 can be rewritten as 1/(x^5) and y^-3 can be rewritten as 1/(y^3).

Our expression now becomes 2 * 1/(x^5) * 1/(y^3).

To simplify further, we can multiply the numerators and the denominators together. So, the expression becomes 2/(x^5 * y^3) or 2/(x^5y^3).

3. To simplify the expression (4x^4y^-3)^-2, we can use the property that raising a power to a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. So, the expression becomes 1 / (4x^4y^-3)^2.

Now, we can square each term inside the parentheses. 4^2 is 16 (since 4 * 4 = 16), x^4 raised to the second power becomes x^8 (since we multiply the exponents), and y^-3 raised to the second power becomes y^-6 (since we multiply the exponents).

Our expression now becomes 1 / (16x^8y^-6).

Using the rule that a negative exponent represents taking the reciprocal, we can rewrite y^-6 as 1/(y^6).

Our expression further simplifies to 1 / (16x^8 * y^6).

4. To simplify the expression 1/(3x)^-2, we can use the property that a negative exponent represents taking the reciprocal. So, the expression becomes (3x)^2.

Expanding, (3x)^2 is equal to (3^2)(x^2) or 9x^2.

Therefore, the simplified expression is 9x^2.