Question

properties of exponents
use the properties of exponents to simplify each expression. match each correct answer to a letter and complete the riddle below.

1. $\frac{x^{12}}{x^{6}}$ 2. $x^{10}cdot x^{2}$
2. $(x^{4})^{5}$ 4. $(xy)^{5}$
3. $\frac{x^{8}cdot x^{4}}{x^{3}}$ 6. $\frac{x^{10}}{x^{2}}cdot x^{6}$
4. $(x^{2}y^{2})^{3}$ 8. $(x^{6})^{7}cdot(x^{3})^{3}cdot x$
5. $\frac{x^{24}}{(x^{4})^{2}}$ 10. $\frac{(xy)^{7}}{(x^{2}y^{3})^{2}}$

i: $x^{9}$ e: $x^{12}$ r: $x^{2}$ d: $x^{18}$ e: $x^{3}y$ s: $x^{14}$
s: $x^{16}$ l: $x^{20}$ e: $x^{6}y^{6}$ o: $x^{5}y^{5}$ a: $x^{15}$ j: $x$
t: $x^{3}$ n: $xy^{5}$ c: $x^{22}$ s: $x^{16}$ u: $x^{3}y^{2}$ c: $x^{6}$
what kinds of triangles are the coldest?
5 8 2 9 4 6 1 7 3 10 9

Explanation:

Step1: Simplify $\frac{x^{12}}{x^{6}}$

Use the quotient - rule $a^m\div a^n=a^{m - n}$, so $\frac{x^{12}}{x^{6}}=x^{12 - 6}=x^{6}$ (C).

Step2: Simplify $x^{10}\cdot x^{2}$

Use the product - rule $a^m\cdot a^n=a^{m + n}$, so $x^{10}\cdot x^{2}=x^{10 + 2}=x^{12}$ (E).

Step3: Simplify $(x^{4})^{5}$

Use the power - of - a - power rule $(a^m)^n=a^{mn}$, so $(x^{4})^{5}=x^{4\times5}=x^{20}$ (L).

Step4: Simplify $(xy)^{5}$

Use the power - of - a - product rule $(ab)^n=a^n b^n$, so $(xy)^{5}=x^{5}y^{5}$ (O).

Step5: Simplify $\frac{x^{8}\cdot x^{4}}{x^{3}}$

First, use the product rule for the numerator: $x^{8}\cdot x^{4}=x^{8 + 4}=x^{12}$. Then use the quotient rule: $\frac{x^{12}}{x^{3}}=x^{12-3}=x^{9}$ (I).

Step6: Simplify $\frac{x^{10}}{x^{2}}\cdot x^{6}$

First, use the quotient rule: $\frac{x^{10}}{x^{2}}=x^{10 - 2}=x^{8}$. Then use the product rule: $x^{8}\cdot x^{6}=x^{8 + 6}=x^{14}$ (S).

Step7: Simplify $(x^{2}y^{2})^{3}$

Use the power - of - a - product rule: $(x^{2}y^{2})^{3}=(x^{2})^{3}(y^{2})^{3}=x^{2\times3}y^{2\times3}=x^{6}y^{6}$ (E).

Step8: Simplify $(x^{6})^{7}\cdot(x^{3})^{3}\cdot x$

Use the power - of - a - power rule: $(x^{6})^{7}=x^{6\times7}=x^{42}$, $(x^{3})^{3}=x^{3\times3}=x^{9}$. Then use the product rule: $x^{42}\cdot x^{9}\cdot x=x^{42 + 9+1}=x^{52}$ (There is no match in the given options, assuming there is a mistake in the problem - setup. Let's re - check the rules application). Re - calculating: $(x^{6})^{7}\cdot(x^{3})^{3}\cdot x=x^{42}\cdot x^{9}\cdot x=x^{42 + 9+1}=x^{52}$. If we assume the intended problem is $(x^{6})^{2}\cdot(x^{3})^{3}\cdot x=x^{12}\cdot x^{9}\cdot x=x^{12 + 9+1}=x^{22}$ (C).

Step9: Simplify $\frac{x^{24}}{(x^{4})^{2}}$

Use the power - of - a - power rule: $(x^{4})^{2}=x^{4\times2}=x^{8}$. Then use the quotient rule: $\frac{x^{24}}{x^{8}}=x^{24 - 8}=x^{16}$ (S).

Step10: Simplify $\frac{(xy)^{7}}{(x^{2}y^{3})^{2}}$

First, use the power - of - a - product rule: $(xy)^{7}=x^{7}y^{7}$ and $(x^{2}y^{3})^{2}=x^{2\times2}y^{3\times2}=x^{4}y^{6}$. Then use the quotient rule: $\frac{x^{7}y^{7}}{x^{4}y^{6}}=x^{7 - 4}y^{7 - 6}=x^{3}y$ (E).

Answer:

1. C: $x^{6}$
2. E: $x^{12}$
3. L: $x^{20}$
4. O: $x^{5}y^{5}$
5. I: $x^{9}$
6. S: $x^{14}$
7. E: $x^{6}y^{6}$
8. C: $x^{22}$
9. S: $x^{16}$
10. E: $x^{3}y$