Question

1. write each expression as a power with positive exponents. state the exponent law(s) you used.

a) \\(x^2 \cdot x^5\\)
b) \\(x^2 \cdot x^{-12}\\)
c) \\((x^{-2})^{-3}\\)
d) \\((x^2 y^{-3})^{-4}\\)
e) \\(\frac{x^4 y^2}{x^2 y^{-3}}\\)

Explanation:

Simplify part a

$$ x^2 \cdot x^3 = x^{2+3} = x^5 $$

Law used: Product of Powers Law, \(a^m \cdot a^n = a^{m+n}\).

Simplify part b

$$ x^2 \cdot x^{-3} = x^{2 + (-3)} = x^{-1} = \frac{1}{x} $$

Laws used: Product of Powers Law, \(a^m \cdot a^n = a^{m+n}\); Negative Exponent Law, \(a^{-n} = \frac{1}{a^n}\).

Simplify parts c, d, and e

For c:

$$ (x^{-1})^3 = x^{-1 \cdot 3} = x^{-3} = \frac{1}{x^3} $$

Laws used: Power of a Power Law, \((a^m)^n = a^{mn}\); Negative Exponent Law, \(a^{-n} = \frac{1}{a^n}\).

For d:

$$ (x^2 y^{-3})^{-1} = x^{2 \cdot (-1)} y^{-3 \cdot (-1)} = x^{-2} y^3 = \frac{y^3}{x^2} $$

Laws used: Power of a Product Law, \((ab)^n = a^n b^n\); Power of a Power Law, \((a^m)^n = a^{mn}\); Negative Exponent Law, \(a^{-n} = \frac{1}{a^n}\).

For e:

$$ \frac{x^2 y^3}{x^{-1} y^2} = x^{2 - (-1)} y^{3 - 2} = x^3 y^1 = x^3 y $$

Law used: Quotient of Powers Law, \(\frac{a^m}{a^n} = a^{m-n}\).

Answer:

Question 9a

Expression: \(x^5\)
Exponent Law(s) used: Product of Powers Law

Question 9b

Expression: \(\frac{1}{x}\)
Exponent Law(s) used: Product of Powers Law, Negative Exponent Law

Question 9c

Expression: \(\frac{1}{x^3}\)
Exponent Law(s) used: Power of a Power Law, Negative Exponent Law

Question 9d

Expression: \(\frac{y^3}{x^2}\)
Exponent Law(s) used: Power of a Product Law, Power of a Power Law, Negative Exponent Law

Question 9e

Expression: \(x^3 y\)
Exponent Law(s) used: Quotient of Powers Law