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.
8^3 x^3/8^4
= x^3/8
2. 2/x^5 * 1/y^3
= 2/(x^5 y^3)
3. 1/(4x^4 y^-3)^2
= 1/(16x^8 y^-6)
= y^6 /(16x^8)
4. 1/(3x)^-2
= (3x)^2
= 9x^2
I don't understand how you got that for #1.
1. Notation confusion maybe
(8^3 x^3) / 8^4
= (1/8) x^3
1. To simplify the expression (8x)^3 * 8^(-4), we need to apply the exponent rules.
First, let's simplify (8x)^3. This means we need to take each term inside the parentheses (8 and x) to the power of 3.
So, (8x)^3 = 8^3 * x^3.
8^3 = 8 * 8 * 8 = 512, and x^3 remains the same.
Now we have 512 * x^3 * 8^(-4).
Next, let's simplify 8^(-4). A negative exponent means we need to take the reciprocal of the base raised to the positive exponent.
So, 8^(-4) = 1/(8^4) = 1/8^4 = 1/4096.
Now we have 512 * x^3 * 1/4096.
To multiply fractions, we multiply the numerators together and the denominators together.
So, 512 * x^3 * 1/4096 = (512 * x^3)/(4096).
To simplify further, we can divide both the numerator and denominator by their greatest common factor, which is 512.
512 divided by 512 is equal to 1, so we can simplify the expression to:
x^3/8.
Therefore, the simplified expression is x^3/8.
2. To simplify the expression 2x^(-5) * y^(-3), we need to apply the exponent rules.
First, let's simplify x^(-5). A negative exponent means we need to take the reciprocal of the base raised to the positive exponent.
So, x^(-5) = 1/(x^5) = 1/x^5.
Similarly, y^(-3) = 1/(y^3) = 1/y^3.
Now we have 2 * 1/x^5 * 1/y^3.
To multiply fractions, multiply the numerators together and the denominators together.
So, 2 * 1/x^5 * 1/y^3 = 2/(x^5 * y^3).
Therefore, the simplified expression is 2/(x^5 * y^3).
3. To simplify the expression (4x^4 * y^(-3))^(-2), we need to apply the exponent rules.
First, let's simplify the term inside the parentheses, 4x^4 * y^(-3).
We can leave 4x^4 as is, but for y^(-3), we need to take the reciprocal of the base raised to the positive exponent.
So, y^(-3) = 1/(y^3) = 1/y^3.
Now we have 4x^4 * 1/y^3.
Next, we raise this entire expression to the power of -2.
To raise a fraction to a negative exponent, we can take the reciprocal of the fraction and change the exponent to positive.
So, (4x^4 * 1/y^3)^(-2) = (y^3/4x^4)^2.
To simplify further, we square both the numerator and denominator.
So, (y^3/4x^4)^2 = (y^3)^2/(4x^4)^2.
Expanding the exponents, we get (y^6)/(16x^8).
Therefore, the simplified expression is y^6/(16x^8).
4. To simplify the expression 1/(3x)^(-2), we need to apply the exponent rules.
First, let's simplify (3x)^(-2). This means we need to take each term inside the parentheses (3 and x) to the power of -2.
So, (3x)^(-2) = 3^(-2) * x^(-2).
A negative exponent means we need to take the reciprocal of the base raised to the positive exponent.
So, 3^(-2) = 1/(3^2) = 1/9, and x^(-2) = 1/(x^2) = 1/x^2.
Now we have 1/9 * 1/x^2.
To multiply these fractions, multiply the numerators together and the denominators together.
So, 1/9 * 1/x^2 = 1/(9x^2).
Therefore, the simplified expression is 1/(9x^2).