Question

negative exponent and zero exponent properties

1. $a^{-7} = $
2. $(21c^{18})^{-1} = $
3. $(3d^2)^0 = $
4. $5(x^0)y^{-1} = $

product of powers property

1. $a^7a^{12} = $
2. $c^3c^8c^{-5} = $
3. $(2d^7)(-4d^9d^5) = $
4. $(9x^{10}y^3)(-x^5y^3) = $

quotient of powers property

1. $\frac{a^{12}}{a^7} = $
2. $\frac{6c^3}{3c^{-5}} = $
3. $\frac{2d^7}{-4d^9d^5} = $
4. $\frac{9x^{10}y^3}{-x^5y^3} = $

power of a power property

1. $(a^3)^4 = $
2. $(c^{-1})^3 = $
3. $(d^5)^{-2} = $
4. $(6x^3y)(x^2)^{-2} = $

Explanation:

Step1: Apply negative exponent rule

$a^{-7} = \frac{1}{a^7}$

Step2: Apply negative exponent rule

$(21c^{18})^{-1} = \frac{1}{21c^{18}}$

Step3: Apply zero exponent rule

$(3d^2)^0 = 1$

Step4: Apply zero/negative exponent rules

$5(x^0)y^{-1} = 5(1)\frac{1}{y} = \frac{5}{y}$

Step5: Add exponents for product

$a^7a^{12} = a^{7+12} = a^{19}$

Step6: Add exponents for product

$c^3c^8c^{-5} = c^{3+8-5} = c^6$

Step7: Multiply coefficients, add exponents

$(2d^7)(-4d^9d^5) = 2(-4)d^{7+9+5} = -8d^{21}$

Step8: Multiply coefficients, add exponents

$(9x^{10}y^3)(-x^5y^3) = 9(-1)x^{10+5}y^{3+3} = -9x^{15}y^6$

Step9: Subtract exponents for quotient

$\frac{a^{12}}{a^7} = a^{12-7} = a^5$

Step10: Divide coefficients, add exponents

$\frac{6c^3}{3c^{-5}} = \frac{6}{3}c^{3-(-5)} = 2c^8$

Step11: Divide coefficients, add exponents

$\frac{2d^7}{-4d^9d^5} = \frac{2}{-4}d^{7-(9+5)} = -\frac{1}{2}d^{-7} = \frac{-1}{2d^7}$

Step12: Divide coefficients, subtract exponents

$\frac{9x^{10}y^3}{-x^5y^3} = \frac{9}{-1}x^{10-5}y^{3-3} = -9x^5$

Step13: Multiply exponents for power of power

$(a^3)^4 = a^{3\times4} = a^{12}$

Step14: Multiply exponents, apply negative rule

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

Step15: Multiply exponents, apply negative rule

$(d^5)^{-2} = d^{-10} = \frac{1}{d^{10}}$

Step16: Multiply exponents, combine terms

$(6x^3y)(x^2)^{-2} = 6x^3y x^{-4} = 6x^{3-4}y = \frac{6y}{x}$

Answer:

1. $\frac{1}{a^7}$
2. $\frac{1}{21c^{18}}$
3. $1$
4. $\frac{5}{y}$
5. $a^{19}$
6. $c^6$
7. $-8d^{21}$
8. $-9x^{15}y^6$
9. $a^5$
10. $2c^8$
11. $\frac{-1}{2d^7}$
12. $-9x^5$
13. $a^{12}$
14. $\frac{1}{c^3}$
15. $\frac{1}{d^{10}}$
16. $\frac{6y}{x}$