I'm really stuck on these problems-
Simplify 6xy(3xy)^-2?
Simplify 12xyz^-2/8x^-2z
Thank you
To simplify the expression 6xy(3xy)^-2, follow these steps:
Step 1: Simplify the exponent inside the parentheses.
Since the exponent is -2, it means that we need to take the reciprocal (inverse) and square the term inside the parentheses.
So, (3xy)^-2 becomes (1/(3xy))^2.
Step 2: Square the term inside the parentheses.
(1/(3xy))^2 = 1^2 / (3xy)^2.
Simplifying further, we have 1 / (3^2 * x^2 * y^2).
This simplifies to 1 / (9x^2y^2).
Step 3: Multiply the simplified term with the term outside the parentheses.
Now, we have 6xy multiplied by 1 / (9x^2y^2).
Multiplying the numerators and denominators together, we get (6xy)/(9x^2y^2).
Step 4: Simplify the expression further.
To simplify the fractional expression, we divide the numerator and denominator by the greatest common factor (GCF) of the terms.
In this case, the GCF is 3xy. Dividing both the numerator and denominator by 3xy, we get:
(6xy)/(9x^2y^2) = (2)/(3xy).
Therefore, the simplified form of 6xy(3xy)^-2 is 2/(3xy).
Moving on to the second problem:
To simplify the expression 12xyz^-2 / 8x^(-2)z, follow these steps:
Step 1: Simplify the negative exponents.
In this expression, we have two negative exponents: y^-2 and x^-2.
To simplify them, we can take their reciprocals and change the signs of the exponents.
So, y^-2 becomes 1/y^2, and x^-2 becomes 1/x^2.
Step 2: Apply the simplified exponents to the expression.
The expression becomes (12xz)/(8z * x^2 * y^2).
Step 3: Simplify the numbers.
12xz divided by 8z is equivalent to 3xz divided by 2z. Canceling out the z terms, we get 3x/2.
Step 4: Simplify the variables.
x divided by x^2 is equivalent to 1/x. Therefore, we have 3/x * 1/y^2.
So, the final simplified form of 12xyz^-2 / 8x^-2z is 3/(2xy^2).
To simplify the expressions, we can apply the following rules:
1. For multiplication and division of exponential expressions with the same base: keep the base the same and add/subtract the exponents.
2. A negative exponent means taking the reciprocal of the base with a positive exponent.
3. If there are parentheses involved, apply the exponent to everything inside the parentheses.
Now, let's solve each problem step-by-step:
Problem 1: Simplify 6xy(3xy)^-2.
Step 1: Simplify the expression inside the parentheses.
(3xy)^-2 = 1 / (3xy)^2
Step 2: Square the expression inside the parentheses.
(3xy)^2 = 9x^2y^2
Step 3: Apply the reciprocal to the squared expression.
1 / (9x^2y^2) = (x^2y^2) / 9
Step 4: Multiply the result from Step 3 by 6xy.
6xy * (x^2y^2 / 9)
Step 5: Simplify the expression further.
(6x * x^2 * y * y^2) / 9 = (6x^3y^3) / 9
The final simplified expression is (6x^3y^3) / 9.
Problem 2: Simplify 12xyz^-2 / 8x^-2z.
Step 1: Apply the reciprocal to the term with a negative exponent.
xyz^-2 = x * y * (1 / z^2) = xy / z^2
Step 2: Multiply by the reciprocal of 8x^-2z.
12xyz^-2 / 8x^-2z = 12xy / (8 * 1 / x^2 * z)
Step 3: Simplify the expression further.
12xy / (8 / x^2 * z) = 12xy * (x^2 * z / 8)
Step 4: Cancel out common factors.
12xy * (x^2 * z / 8) = 3xy * (x * z / 2)
The final simplified expression is 3xy * (xz / 2).
I hope this explanation helps you understand how to simplify these expressions! If you have any further questions, feel free to ask.