Question

1. reinforce create expressions that meet the conditions specified.

a. the expression simplifies to ( 4^{2}x^{4}y ) using the multiplication rule.
b. the expression simplifies to ( \frac{5x^{7}}{y^{3}} ) using the division rule, and has at least one negative exponent.
c. the expression simplifies to ( 5^{3}x^{5}y^{2} ) using the multiplication rule and the power rule, and has at least two negative exponents.

Explanation:

Part a

Step1: Recall the multiplication rule of exponents \(a^m\times a^n=a^{m + n}\)

We can write \(4x^2y\times4x^2=4\times4\times x^{2+2}\times y = 4^{2}x^{4}y\)

Part b

Step1: Recall the division rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\)

We know that \(\frac{5x^{10}}{y^{4}}=5x^{10-3}y^{0 - 3}=5x^{7}y^{- 3}=\frac{5x^{7}}{y^{3}}\) (using \(a^{-n}=\frac{1}{a^{n}}\))

Part c

Step1: Recall the multiplication rule \(a^m\times a^n=a^{m + n}\) and power rule \((a^m)^n=a^{mn}\)

Let's consider \(\frac{5^{5}x^{7}y^{4}}{5^{2}x^{2}y^{2}}=5^{5 - 2}x^{7-2}y^{4 - 2}=5^{3}x^{5}y^{2}\). We can also write it using negative exponents: \(5^{5}x^{7}y^{4}\times5^{-2}x^{-2}y^{-2}=5^{5+( - 2)}x^{7+( - 2)}y^{4+( - 2)}=5^{3}x^{5}y^{2}\)

Answer:

a. \(4x^2y\times4x^2\)
b. \(\frac{5x^{10}}{y^{4}}\)
c. \(5^{5}x^{7}y^{4}\times5^{-2}x^{-2}y^{-2}\)