Question

properties of exponents
product rule
directions: circle the correct answer in each row.
question a b c d

1. simplify the product below.

5^4 • 5^6 5^24 5^10 25^10 25^24

1. simplify the product below.

x^7 • x^9 x^2 x^63 x^79 x^16

1. simplify the product below.

b^3 • b^11 b^33 b^14 b^8 2b^2

1. simplify the product below.

x^4 • x^2 • x^3 x^11 x^24 x^9 x^10

1. simplify the product below.

y^2 • y^3 • y^4 • y^5 y^4 y^13 y^14 y^15

1. simplify the product below.

h^2 • h^6 • h h^13 h^9 h^8 h^4

1. simplify the product below.

7^11 • 7^8 7^3 49^19 7^19 49^3

1. simplify the product below.

(x^3)(xy^2) x^2y x^3y^2 x^4y^2 x^4y^3

1. simplify the product below.

(x^2 y^4)(xy^3) x^2y^1 x^2y^7 x^3y x^3y^7

1. simplify the product below.

(2ab)(ab)(a^2b^4) 2a^2b^4 2a^4b^6 -2a^2b^4 -2a^4b^6

1. simplify the product below.

x(x^3 + x) x^3 + x x^4 + x x^4 + x^2 x^4 + 2x

1. find the value of a:

x^5 • x^a = x^8 a = 3 a = 5 a = 8 a = 13

1. find the value of a:

y^3 • y^5 • y^a = y^15 a = 3 a = 7 a = 8 a = 13

1. find the product:

a(3a^2 - 4a - 8) 3a^3-4a-8 3a^3 - 4a^2-8a 3a^2 -4a -8 3a^2 -4a -8

1. find the product:

(x^2 y^3 z^6)(x^5y^9z^3) x^7y^12z^9 x^25y^39z^43 x^7y^12z^9 x^10y^27z^18

Explanation:

Step1: Recall product - rule of exponents

The product - rule states that $a^m\cdot a^n=a^{m + n}$ for the same base $a$.

Step2: Solve question 1

For $5^4\cdot5^6$, using the product - rule $a^m\cdot a^n=a^{m + n}$ with $a = 5$, $m = 4$, and $n = 6$, we get $5^{4+6}=5^{10}$.

Step3: Solve question 2

For $x^7\cdot x^9$, using the product - rule with $a=x$, $m = 7$, and $n = 9$, we have $x^{7 + 9}=x^{16}$.

Step4: Solve question 3

For $b^3\cdot b^{11}$, using the product - rule with $a = b$, $m = 3$, and $n = 11$, we get $b^{3+11}=b^{14}$.

Step5: Solve question 4

For $x^4\cdot x^2\cdot x^3$, using the product - rule multiple times: $x^{4+2+3}=x^9$.

Step6: Solve question 5

For $y^2\cdot y^3\cdot y^4\cdot y^5$, we have $y^{2 + 3+4+5}=y^{14}$.

Step7: Solve question 6

For $h^2\cdot h^6\cdot h=h^2\cdot h^6\cdot h^1$, then $h^{2+6 + 1}=h^9$.

Step8: Solve question 7

For $7^{11}\cdot7^8$, using the product - rule with $a = 7$, $m = 11$, and $n = 8$, we get $7^{11+8}=7^{19}$.

Step9: Solve question 8

For $(x^3)(xy^2)$, we can rewrite it as $x^3\cdot x^1\cdot y^2$. Using the product - rule for $x$ terms: $x^{3+1}\cdot y^2=x^4y^2$.

Step10: Solve question 9

For $(x^2y^4)(xy^3)$, we have $x^2\cdot x^1\cdot y^4\cdot y^3$. Using the product - rule for $x$ and $y$ terms: $x^{2 + 1}y^{4+3}=x^3y^7$.

Step11: Solve question 10

For $(2ab)(ab)(a^2b^4)$, we first multiply the coefficients ($2\times1\times1 = 2$) and then use the product - rule for $a$ and $b$ terms. For $a$: $a^1\cdot a^1\cdot a^2=a^{1 + 1+2}=a^4$, for $b$: $b^1\cdot b^1\cdot b^4=b^{1+1 + 4}=b^6$. So the result is $2a^4b^6$.

Step12: Solve question 11

For $x(x^3 + x)=x\cdot x^3+x\cdot x$. Using the product - rule, $x^{1+3}+x^{1 + 1}=x^4+x^2$.

Step13: Solve question 12

Given $x^5\cdot x^a=x^8$. Using the product - rule $x^{5 + a}=x^8$. Then $5 + a=8$, so $a=8 - 5=3$.

Step14: Solve question 13

Given $y^3\cdot y^5\cdot y^a=y^{15}$. First, $y^3\cdot y^5=y^{3 + 5}=y^8$. Then $y^8\cdot y^a=y^{8 + a}=y^{15}$. So $8 + a=15$, and $a=15 - 8 = 7$.

Step15: Solve question 14

For $a(3a^2-4a - 8)=a\cdot3a^2-a\cdot4a-a\cdot8$. Using the product - rule: $3a^{1+2}-4a^{1 + 1}-8a=3a^3-4a^2-8a$.

Step16: Solve question 15

For $(x^2y^3z^6)(x^5y^9z^3)$, using the product - rule for $x$, $y$, and $z$ terms: $x^{2+5}y^{3 + 9}z^{6+3}=x^7y^{12}z^9$.

Answer:

1. B. $5^{10}$
2. D. $x^{16}$
3. B. $b^{14}$
4. C. $x^9$
5. C. $y^{14}$
6. B. $h^9$
7. C. $7^{19}$
8. C. $x^4y^2$
9. D. $x^3y^7$
10. B. $2a^4b^6$
11. C. $x^4+x^2$
12. A. $a = 3$
13. B. $a = 7$
14. B. $3a^3-4a^2-8a$
15. C. $x^7y^{12}z^9$