Hi! How do you simplify the following using the laws of exponents?
1. -2m^2 n^-3 /(mn^-2)^2
2. (-2pq)^-2 / 2pq^-2
For number 1, I don't have any idea but for number 2, I arrived at 1/2p. Is that right? Thanks much in advance.
#1.
-2m^2 n^-3 /(mn^-2)^2
-2m^2 n^-3 /(m^2n^-4)
-2m^(2-2) n^(-3+4)
-2n
#2.
(-2pq)^-2 / 2pq^-2
(-2)^-2 p^-2 q^-1 / 2pq^-2
1/8 p^(-2-1) q^(-1+2)
q/8p^3
Oops.
#2.
(-2pq)^-2 / 2pq^-2
(-2)^-2 p^-2 q^-2 / 2pq^-2
1/8 p^(-2-1) q^(-2+2)
1/8p^3
Hi! I'm here to help you. Let's simplify each expression using the laws of exponents.
For number 1: -2m^2 n^-3 /(mn^-2)^2
Step 1: Start by simplifying the expression inside the parentheses: mn^-2. Remember that multiplying variables with the same base involves adding the exponents.
mn^-2 = m * n^(-2) = m/n^2
Step 2: Substitute this simplified expression back into the original expression.
-2m^2 n^-3 /(mn^-2)^2 = -2m^2 n^-3 /(m/n^2)^2
Step 3: Evaluate the power of a quotient rule. When you raise a quotient to a power, you can distribute the exponent to both the numerator and denominator.
-2m^2 n^-3 /(m/n^2)^2 = -2m^2 n^-3 * n^4 / m^2
= -2m^2 * n^-3 * n^4 / m^2
Step 4: Apply the rule of multiplying exponents. When you multiply variables with the same base, you can add their exponents.
-2m^2 * n^-3 * n^4 / m^2 = -2m^2 * n^(4-3) / m^2
= -2m^2 * n^1 / m^2
= -2mn / m^2
Step 5: Simplify further by canceling out common factors.
-2mn / m^2 = -2n / m
So, the simplified form is -2n / m.
For number 2: (-2pq)^-2 / 2pq^-2
Step 1: Start by simplifying the expression inside the parentheses: -2pq.
(-2pq)^-2 = (-2)^-2 * (pq)^-2
= 1/(-2)^2 * (pq)^-2
= 1/4 * (pq)^-2
Step 2: Evaluate the negative exponent rule. A negative exponent indicates taking the reciprocal of the base raised to the positive exponent.
1/4 * (pq)^-2 = 1/4 * 1/(pq)^2
Step 3: Apply the rule of raising a power to a power. When you raise a power to another power, you multiply the exponents.
1/4 * 1/(pq)^2 = 1/4 * 1/p^2q^2
Step 4: Combine the fractions by multiplying the numerators and the denominators.
1/4 * 1/p^2q^2 = 1 / (4p^2q^2)
So, the simplified form is 1 / (4p^2q^2).
Based on your explanation for number 2, your answer of 1/2p is incorrect. The correct simplified form is 1 / (4p^2q^2).