Question

simplify using the properties of exponents. circle your final answers.
1.) \\(\frac{42}{63}\\) 2.) \\(\frac{10^4}{10^6}\\) 3.) \\(\frac{12a}{3a}\\)

4.) \\(\frac{6\cdot10^5}{3\cdot10^3}\\) 5.) \\(\frac{4x^5}{8x}\\) 6.) \\(\frac{12y^3}{4xy}\\)

7.) \\(\frac{-9x^2y^3}{12xy^2}\\) 8.) \\(\frac{-28cd^3}{-21bd^2}\\) 9.) \\(\frac{xy^2z^3}{x^3y^2z}\\)

10.) \\(\frac{32a^2bc^3}{20abc}\\) 11.) \\(\frac{(2r)^4}{2r}\\) 12.) \\(\frac{(3x^2)^3}{(3x^3)^2}\\)

13.) \\(\frac{(2ab)^2}{2ab^2}\\) 14.) \\(\frac{(-x)^9}{(-x)^6}\\) 15.) \\(\frac{(-xy)^9}{xy^9}\\)

Explanation:

Step1: Simplify fraction by GCD

$\frac{42}{63} = \frac{42\div21}{63\div21} = \frac{2}{3}$

Step2: Apply exponent rule $\frac{a^m}{a^n}=a^{m-n}$

$\frac{10^4}{10^6} = 10^{4-6} = 10^{-2} = \frac{1}{100}$

Step3: Cancel common factor $a$

$\frac{12a}{3a} = \frac{12}{3} = 4$

Step4: Split and simplify terms

$\frac{6\cdot10^5}{3\cdot10^3} = \frac{6}{3} \cdot 10^{5-3} = 2 \cdot 10^2 = 200$

Step5: Cancel $x$ and simplify coefficient

$\frac{4x^5}{8x} = \frac{4}{8} \cdot x^{5-1} = \frac{1}{2}x^4$

Step6: Cancel common factors

$\frac{12y^3}{4xy} = \frac{12}{4} \cdot \frac{y^3}{y} \cdot \frac{1}{x} = 3 \cdot y^{2} \cdot \frac{1}{x} = \frac{3y^2}{x}$

Step7: Cancel and simplify terms

$\frac{-9x^2y^3}{12xy^2} = \frac{-9}{12} \cdot x^{2-1} \cdot y^{3-2} = -\frac{3}{4}xy$

Step8: Cancel negatives and simplify

$\frac{-28cd^3}{-21bd^2} = \frac{28}{21} \cdot \frac{d^3}{d^2} \cdot \frac{c}{b} = \frac{4}{3} \cdot d \cdot \frac{c}{b} = \frac{4cd}{3b}$

Step9: Apply exponent rules to variables

$\frac{xy^2z^3}{x^3y^2z} = x^{1-3} \cdot y^{2-2} \cdot z^{3-1} = x^{-2} \cdot 1 \cdot z^2 = \frac{z^2}{x^2}$

Step10: Cancel common factors

$\frac{32a^2bc^3}{20abc} = \frac{32}{20} \cdot a^{2-1} \cdot b^{1-1} \cdot c^{3-1} = \frac{8}{5} \cdot a \cdot 1 \cdot c^2 = \frac{8ac^2}{5}$

Step11: Expand numerator then simplify

$\frac{(2r)^4}{2r} = \frac{16r^4}{2r} = 8r^{4-1} = 8r^3$

Step12: Expand exponents then simplify

$\frac{(3x^2)^3}{(3x^3)^2} = \frac{27x^6}{9x^6} = \frac{27}{9} \cdot x^{6-6} = 3 \cdot 1 = 3$

Step13: Expand numerator then simplify

$\frac{(2ab)^2}{2ab^2} = \frac{4a^2b^2}{2ab^2} = \frac{4}{2} \cdot a^{2-1} \cdot b^{2-2} = 2a \cdot 1 = 2a$

Step14: Apply exponent rule $\frac{a^m}{a^n}=a^{m-n}$

$\frac{(-x)^9}{(-x)^6} = (-x)^{9-6} = (-x)^3 = -x^3$

Step15: Expand numerator then simplify

$\frac{(-xy)^9}{xy^9} = \frac{-x^9y^9}{xy^9} = -x^{9-1} \cdot y^{9-9} = -x^8 \cdot 1 = -x^8$

Answer:

1.) $\frac{2}{3}$
2.) $\frac{1}{100}$ or $10^{-2}$
3.) $4$
4.) $200$
5.) $\frac{1}{2}x^4$
6.) $\frac{3y^2}{x}$
7.) $-\frac{3}{4}xy$
8.) $\frac{4cd}{3b}$
9.) $\frac{z^2}{x^2}$
10.) $\frac{8ac^2}{5}$
11.) $8r^3$
12.) $3$
13.) $2a$
14.) $-x^3$
15.) $-x^8$