Question

simplify using the properties of exponents. circle your final answer.
1.) $m^4 \cdot m^5$
2.) $\left(m^2\
ight)^7$
3.) $\left(- 6xy^4\
ight)\left(\frac{1}{2}xy^7\
ight)$
4.) $\frac{2a^6b^3}{5} \cdot \frac{25a^4b}{6}$
5.) $\left(- 5x^5y^4\
ight)^3$
6.) $(- x)^6$
7.) $\left(a^2b^3\
ight)^4\left(3ab^4\
ight)^3$
8.) $\left(- 2x^2\
ight)^5$
9.) $\left(5c^2\
ight)(- d)\left(- \frac{1}{5}cd^2\
ight)$
10.) $\left(\frac{5}{2}m^5\
ight)\left(\frac{10}{20}m^7\
ight)$
11.) $\left(90a^5\
ight)\left(\frac{1}{3}a^4\
ight)^2$
12.) $(- x)^5(- 3x^2)$
13.) $\left(6x^4\
ight)\left(- 2x^5\
ight) - \left(3x^3\
ight)\left(- x^6\
ight)$
14.) $\left(7x^2\
ight)\left(- 4x^8\
ight) + \left(- 2x^4\
ight)\left(- x^6\
ight)$

Explanation:

Step1: Apply product rule ($a^m \cdot a^n=a^{m+n}$)

$m^{4+5}=m^9$

Step2: Apply power rule ($(a^m)^n=a^{m \cdot n}$)

$(m^2)^7=m^{2 \cdot 7}=m^{14}$

Step3: Multiply coefficients, add exponents

$(-6 \cdot \frac{1}{2})x^{1+1}y^{4+7}=-3x^2y^{11}$

Step4: Multiply fractions, add exponents

$\frac{2 \cdot 25}{5 \cdot 6}a^{6+4}b^{3+1}=\frac{50}{30}a^{10}b^4=\frac{5}{3}a^{10}b^4$

Step5: Distribute power to all terms

$(-5)^3x^{5 \cdot 3}y^{4 \cdot 3}=-125x^{15}y^{12}$

Step6: Even exponent makes result positive

$(-x)^6=x^6$

Step7: Expand powers, then multiply

$(a^{2 \cdot 4}b^{3 \cdot 4})(3^3a^{1 \cdot 3}b^{4 \cdot 3})=(a^8b^{12})(27a^3b^{12})=27a^{8+3}b^{12+12}=27a^{11}b^{24}$

Step8: Distribute power to all terms

$(-2)^5x^{2 \cdot 5}=-32x^{10}$

Step9: Multiply coefficients, add exponents

$(5 \cdot -1 \cdot -\frac{1}{5})c^{2+1}d^{1+2}=1 \cdot c^3d^3=c^3d^3$

Step10: Multiply fractions, add exponents

$\frac{5 \cdot 10}{2 \cdot 20}m^{5+7}=\frac{50}{40}m^{12}=\frac{5}{4}m^{12}$

Step11: Expand power, then multiply

$90a^5 \cdot (\frac{1}{3})^2a^{4 \cdot 2}=90a^5 \cdot \frac{1}{9}a^8=10a^{5+8}=10a^{13}$

Step12: Multiply coefficients, add exponents

$(-1 \cdot -3)x^{5+2}=3x^7$

Step13: Compute products, then subtract

$(6 \cdot -2)x^{4+5} - (3 \cdot -1)x^{3+6}=-12x^9 + 3x^9=-9x^9$

Step14: Compute products, then add

$(7 \cdot -4)x^{2+8} + (-2 \cdot -1)x^{4+6}=-28x^{10} + 2x^{10}=-26x^{10}$

Answer:

1. $m^9$
2. $m^{14}$
3. $-3x^2y^{11}$
4. $\frac{5}{3}a^{10}b^4$
5. $-125x^{15}y^{12}$
6. $x^6$
7. $27a^{11}b^{24}$
8. $-32x^{10}$
9. $c^3d^3$
10. $\frac{5}{4}m^{12}$
11. $10a^{13}$
12. $3x^7$
13. $-9x^9$
14. $-26x^{10}$