QuestionSimplify these exponential expressions: $x^{-2} y$ , $x y^{-3}$ , $x^{0} y^{5}$ , $x^{3} \cdot x^{7}$ , $x^{7} y^{0}$ , $x^{-5} \cdot x^{10}$ , $x^{11} \cdot x^{5}$ , $\left(x^{3}\right)^{7}$ , $x^{-6} \cdot x^{12}$ , $\left(x^{-5}\right)^{3}$ , $\left(x^{11}\right)^{5}$ , $\frac{x^{14}}{x^{7}}$ , $\left(x^{-6}\right)^{4}$ , $\frac{x^{14}}{x^{-7}}$ , $\frac{x^{30}}{x^{10}}$ , $\left(8 x^{3}\right)^{2}$ , $\frac{x^{30}}{x^{-10}}$ , $\left(-\frac{4}{x}\right)^{3}$ , $\left(6 x^{4}\right)^{2}$ , $\left(-3 x^{2} y^{5}\right)^{2}$ , $\left(-\frac{6}{y}\right)^{3}$ , $\left(3 x^{4}\right)\left(2 x^{7}\right)$ , $\left(-3 x^{4} y^{6}\right)^{3}$ , $\left(-9 x^{3} y\right)\left(-2 x^{6} y^{4}\right)$ , $\left(11 x^{5}\right)\left(9 x^{12}\right)$ , $\frac{8 x^{20}}{2 x^{4}}$ , $\left(-5 x^{4} y\right)\left(-6 x^{7} y^{11}\right)$ , $\frac{25 a^{13} b^{4}}{-5 a^{2} b^{3}}$ , $\frac{20 x^{24}}{10 x^{6}}$ , $\frac{14 b^{7}}{7 b^{14}}$ , $\frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}}$ , $\frac{20 b^{10}}{10 b^{20}}$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with exponential expressions. . The base and exponent rules apply.
3. The base rule states that if you multiply powers with the same base, you add the exponents.
4. The exponent rule states that if you raise a power to a power, you multiply the exponents.
5. Any number raised to the power of0 is1.
6. Any number raised to the power of -n is1 divided by that number raised to the power of n.

STEP 2

Let's start with the first expression $x^{-2} y$
This expression can be simplified by applying the rule for negative exponents.
$x^{-2} y = \frac{y}{x^{2}}$

STEP 3

Next, let's simplify the second expression $x y^{-3}$ This expression can be simplified by applying the rule for negative exponents.
$x y^{-3} = \frac{x}{y^{3}}$

STEP 4

Now, let's simplify the third expression $x^{0} y^{}$ This expression can be simplified by applying the rule for zero exponents.
$x^{0} y^{} =1 \cdot y^{} = y^{}$

STEP 5

Next, let's simplify the fourth expression $x^{3} \cdot x^{7}$ This expression can be simplified by applying the base rule.
$x^{3} \cdot x^{7} = x^{3+7} = x^{10}$

STEP 6

Now, let's simplify the fifth expression $x^{} y^{0}$ This expression can be simplified by applying the rule for zero exponents.
$x^{} y^{0} = x^{} \cdot1 = x^{}$

STEP 7

Next, let's simplify the sixth expression $x^{-5} \cdot x^{10}$ This expression can be simplified by applying the base rule.
$x^{-5} \cdot x^{10} = x^{-5+10} = x^{5}$

STEP 8

Now, let's simplify the seventh expression $x^{11} \cdot x^{5}$ This expression can be simplified by applying the base rule.
$x^{11} \cdot x^{5} = x^{11+5} = x^{16}$

STEP 9

Next, let's simplify the eighth expression $\left(x^{3}\right)^{7}$ This expression can be simplified by applying the exponent rule.
$\left(x^{3}\right)^{7} = x^{3 \cdot7} = x^{21}$

STEP 10

Now, let's simplify the ninth expression $x^{-6} \cdot x^{12}$ This expression can be simplified by applying the base rule.
$x^{-6} \cdot x^{12} = x^{-6+12} = x^{6}$

STEP 11

Next, let's simplify the tenth expression $\left(x^{-5}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(x^{-5}\right)^{3} = x^{-5 \cdot3} = x^{-15}$

STEP 12

Now, let's simplify the eleventh expression $\left(x^{11}\right)^{5}$ This expression can be simplified by applying the exponent rule.
$\left(x^{11}\right)^{5} = x^{11 \cdot5} = x^{55}$

STEP 13

Next, let's simplify the twelfth expression $\frac{x^{}}{x^{7}}$
This expression can be simplified by applying the base rule.
$\frac{x^{}}{x^{7}} = x^{-7} = x^{7}$

STEP 14

Now, let's simplify the thirteenth expression $\left(x^{-6}\right)^{4}$ This expression can be simplified by applying the exponent rule.
$\left(x^{-6}\right)^{4} = x^{-6 \cdot4} = x^{-24}$

STEP 15

Next, let's simplify the fourteenth expression $\frac{x^{14}}{x^{-7}}$
This expression can be simplified by applying the base rule.
$\frac{x^{14}}{x^{-7}} = x^{14-(-7)} = x^{21}$

STEP 16

Now, let's simplify the fifteenth expression $\frac{x^{30}}{x^{10}}$
This expression can be simplified by applying the base rule.
$\frac{x^{30}}{x^{10}} = x^{30-10} = x^{20}$

STEP 17

Next, let's simplify the sixteenth expression $\left( x^{3}\right)^{2}$ This expression can be simplified by applying the exponent rule.
$\left( x^{3}\right)^{2} =^{2} \cdot x^{3 \cdot2} =64x^{6}$

STEP 18

Now, let's simplify the seventeenth expression $\frac{x^{30}}{x^{-10}}$
This expression can be simplified by applying the base rule.
$\frac{x^{30}}{x^{-10}} = x^{30-(-10)} = x^{40}$

STEP 19

Next, let's simplify the eighteenth expression $\left(-\frac{4}{x}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(-\frac{4}{x}\right)^{3} = -64x^{-3}$

STEP 20

Now, let's simplify the nineteenth expression $\left(6 x^{4}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(6 x^{4}\right)^{} =36x^{8}$

STEP 21

Next, let's simplify the twentieth expression $\left(-3 x^{} y^{5}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(-3 x^{} y^{5}\right)^{} =9x^{4}y^{10}$

STEP 22

Now, let's simplify the twenty-first expression $\left(-\frac{6}{y}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(-\frac{6}{y}\right)^{} = -216y^{-}$

STEP 23

Next, let's simplify the twenty-second expression $\left(3 x^{}\right)\left( x^{7}\right)$ This expression can be simplified by applying the base rule.
$\left(3 x^{}\right)\left( x^{7}\right) =6x^{11}$

STEP 24

Now, let's simplify the twenty-third expression $\left(-3 x^{4} y^{6}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(-3 x^{4} y^{6}\right)^{3} = -27x^{12}y^{18}$

STEP 25

Next, let's simplify the twenty-fourth expression $\left(-9 x^{3} y\right)\left(- x^{} y^{4}\right)$ This expression can be simplified by applying the base rule.
$\left(-9 x^{3} y\right)\left(- x^{} y^{4}\right) =18x^{9}y^{5}$

STEP 26

Now, let's simplify the twenty-fifth expression $\left(11 x^{5}\right)\left(9 x^{12}\right)$ This expression can be simplified by applying the base rule.
$\left(11 x^{5}\right)\left(9 x^{12}\right) =99x^{17}$

STEP 27

Next, let's simplify the twenty-sixth expression $\frac{ x^{20}}{ x^{4}}$
This expression can be simplified by applying the base rule.
$\frac{ x^{20}}{ x^{4}} =4x^{16}$

STEP 28

Now, let's simplify the twenty-seventh expression $\left(-5 x^{4} y\right)\left(-6 x^{7} y^{11}\right)$ This expression can be simplified by applying the base rule.
$\left(-5 x^{4} y\right)\left(-6 x^{7} y^{11}\right) =30x^{11}y^{12}$

STEP 29

Next, let's simplify the twenty-eighth expression $\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}}$
This expression can be simplified by applying the base rule.
$\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}} = -5a^{11}b$

STEP 30

Now, let's simplify the twenty-ninth expression $\frac{20 x^{24}}{10 x^{6}}$
This expression can be simplified by applying the base rule.
$\frac{20 x^{24}}{10 x^{6}} =2x^{18}$

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Studdy Solution

STEP 1

STEP 2

Let's start with the first expression x−2yx^{-2} yx−2yThis expression can be simplified by applying the rule for negative exponents.x−2y=yx2x^{-2} y = \frac{y}{x^{2}}x−2y=x2y​

STEP 3

Next, let's simplify the second expression xy−3x y^{-3}xy−3This expression can be simplified by applying the rule for negative exponents.xy−3=xy3x y^{-3} = \frac{x}{y^{3}}xy−3=y3x​

STEP 4

Now, let's simplify the third expression x0yx^{0} y^{}x0yThis expression can be simplified by applying the rule for zero exponents.x0y=1⋅y=yx^{0} y^{} =1 \cdot y^{} = y^{}x0y=1⋅y=y

STEP 5

Next, let's simplify the fourth expression x3⋅x7x^{3} \cdot x^{7}x3⋅x7This expression can be simplified by applying the base rule.x3⋅x7=x3+7=x10x^{3} \cdot x^{7} = x^{3+7} = x^{10}x3⋅x7=x3+7=x10

STEP 6

Now, let's simplify the fifth expression xy0x^{} y^{0}xy0This expression can be simplified by applying the rule for zero exponents.xy0=x⋅1=xx^{} y^{0} = x^{} \cdot1 = x^{}xy0=x⋅1=x

STEP 7

Next, let's simplify the sixth expression x−5⋅x10x^{-5} \cdot x^{10}x−5⋅x10This expression can be simplified by applying the base rule.x−5⋅x10=x−5+10=x5x^{-5} \cdot x^{10} = x^{-5+10} = x^{5}x−5⋅x10=x−5+10=x5

STEP 8

Now, let's simplify the seventh expression x11⋅x5x^{11} \cdot x^{5}x11⋅x5This expression can be simplified by applying the base rule.x11⋅x5=x11+5=x16x^{11} \cdot x^{5} = x^{11+5} = x^{16}x11⋅x5=x11+5=x16

STEP 9

Next, let's simplify the eighth expression (x3)7\left(x^{3}\right)^{7}(x3)7This expression can be simplified by applying the exponent rule.(x3)7=x3⋅7=x21\left(x^{3}\right)^{7} = x^{3 \cdot7} = x^{21}(x3)7=x3⋅7=x21

STEP 10

Now, let's simplify the ninth expression x−6⋅x12x^{-6} \cdot x^{12}x−6⋅x12This expression can be simplified by applying the base rule.x−6⋅x12=x−6+12=x6x^{-6} \cdot x^{12} = x^{-6+12} = x^{6}x−6⋅x12=x−6+12=x6

STEP 11

Next, let's simplify the tenth expression (x−5)3\left(x^{-5}\right)^{3}(x−5)3This expression can be simplified by applying the exponent rule.(x−5)3=x−5⋅3=x−15\left(x^{-5}\right)^{3} = x^{-5 \cdot3} = x^{-15}(x−5)3=x−5⋅3=x−15

STEP 12

Now, let's simplify the eleventh expression (x11)5\left(x^{11}\right)^{5}(x11)5This expression can be simplified by applying the exponent rule.(x11)5=x11⋅5=x55\left(x^{11}\right)^{5} = x^{11 \cdot5} = x^{55}(x11)5=x11⋅5=x55

STEP 13

Next, let's simplify the twelfth expression xx7\frac{x^{}}{x^{7}}x7x​This expression can be simplified by applying the base rule.xx7=x−7=x7\frac{x^{}}{x^{7}} = x^{-7} = x^{7}x7x​=x−7=x7

STEP 14

Now, let's simplify the thirteenth expression (x−6)4\left(x^{-6}\right)^{4}(x−6)4This expression can be simplified by applying the exponent rule.(x−6)4=x−6⋅4=x−24\left(x^{-6}\right)^{4} = x^{-6 \cdot4} = x^{-24}(x−6)4=x−6⋅4=x−24

STEP 15

Next, let's simplify the fourteenth expression x14x−7\frac{x^{14}}{x^{-7}}x−7x14​This expression can be simplified by applying the base rule.x14x−7=x14−(−7)=x21\frac{x^{14}}{x^{-7}} = x^{14-(-7)} = x^{21}x−7x14​=x14−(−7)=x21

STEP 16

Now, let's simplify the fifteenth expression x30x10\frac{x^{30}}{x^{10}}x10x30​This expression can be simplified by applying the base rule.x30x10=x30−10=x20\frac{x^{30}}{x^{10}} = x^{30-10} = x^{20}x10x30​=x30−10=x20

STEP 17

Next, let's simplify the sixteenth expression (x3)2\left( x^{3}\right)^{2}(x3)2This expression can be simplified by applying the exponent rule.(x3)2=2⋅x3⋅2=64x6\left( x^{3}\right)^{2} =^{2} \cdot x^{3 \cdot2} =64x^{6}(x3)2=2⋅x3⋅2=64x6

STEP 18

Now, let's simplify the seventeenth expression x30x−10\frac{x^{30}}{x^{-10}}x−10x30​This expression can be simplified by applying the base rule.x30x−10=x30−(−10)=x40\frac{x^{30}}{x^{-10}} = x^{30-(-10)} = x^{40}x−10x30​=x30−(−10)=x40

STEP 19

Next, let's simplify the eighteenth expression (−4x)3\left(-\frac{4}{x}\right)^{3}(−x4​)3This expression can be simplified by applying the exponent rule.(−4x)3=−64x−3\left(-\frac{4}{x}\right)^{3} = -64x^{-3}(−x4​)3=−64x−3

STEP 20

Now, let's simplify the nineteenth expression (6x4)\left(6 x^{4}\right)^{}(6x4)This expression can be simplified by applying the exponent rule.(6x4)=36x8\left(6 x^{4}\right)^{} =36x^{8}(6x4)=36x8

STEP 21

Next, let's simplify the twentieth expression (−3xy5)\left(-3 x^{} y^{5}\right)^{}(−3xy5)This expression can be simplified by applying the exponent rule.(−3xy5)=9x4y10\left(-3 x^{} y^{5}\right)^{} =9x^{4}y^{10}(−3xy5)=9x4y10

STEP 22

Now, let's simplify the twenty-first expression (−6y)\left(-\frac{6}{y}\right)^{}(−y6​)This expression can be simplified by applying the exponent rule.(−6y)=−216y−\left(-\frac{6}{y}\right)^{} = -216y^{-}(−y6​)=−216y−

STEP 23

Next, let's simplify the twenty-second expression (3x)(x7)\left(3 x^{}\right)\left( x^{7}\right)(3x)(x7)This expression can be simplified by applying the base rule.(3x)(x7)=6x11\left(3 x^{}\right)\left( x^{7}\right) =6x^{11}(3x)(x7)=6x11

STEP 24

Now, let's simplify the twenty-third expression (−3x4y6)3\left(-3 x^{4} y^{6}\right)^{3}(−3x4y6)3This expression can be simplified by applying the exponent rule.(−3x4y6)3=−27x12y18\left(-3 x^{4} y^{6}\right)^{3} = -27x^{12}y^{18}(−3x4y6)3=−27x12y18

STEP 25

STEP 26

Now, let's simplify the twenty-fifth expression (11x5)(9x12)\left(11 x^{5}\right)\left(9 x^{12}\right)(11x5)(9x12)This expression can be simplified by applying the base rule.(11x5)(9x12)=99x17\left(11 x^{5}\right)\left(9 x^{12}\right) =99x^{17}(11x5)(9x12)=99x17

STEP 27

Next, let's simplify the twenty-sixth expression x20x4\frac{ x^{20}}{ x^{4}}x4x20​This expression can be simplified by applying the base rule.x20x4=4x16\frac{ x^{20}}{ x^{4}} =4x^{16}x4x20​=4x16

STEP 28

STEP 29

Next, let's simplify the twenty-eighth expression 25a13b4−5a2b\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}}−5a2b25a13b4​This expression can be simplified by applying the base rule.25a13b4−5a2b=−5a11b\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}} = -5a^{11}b−5a2b25a13b4​=−5a11b

STEP 30

Now, let's simplify the twenty-ninth expression 20x2410x6\frac{20 x^{24}}{10 x^{6}}10x620x24​This expression can be simplified by applying the base rule.20x2410x6=2x18\frac{20 x^{24}}{10 x^{6}} =2x^{18}10x620x24​=2x18

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Let's start with the first expression $x^{-2} y$
This expression can be simplified by applying the rule for negative exponents.
$x^{-2} y = \frac{y}{x^{2}}$

Next, let's simplify the second expression $x y^{-3}$ This expression can be simplified by applying the rule for negative exponents.
$x y^{-3} = \frac{x}{y^{3}}$

Now, let's simplify the third expression $x^{0} y^{}$ This expression can be simplified by applying the rule for zero exponents.
$x^{0} y^{} =1 \cdot y^{} = y^{}$

Next, let's simplify the fourth expression $x^{3} \cdot x^{7}$ This expression can be simplified by applying the base rule.
$x^{3} \cdot x^{7} = x^{3+7} = x^{10}$

Now, let's simplify the fifth expression $x^{} y^{0}$ This expression can be simplified by applying the rule for zero exponents.
$x^{} y^{0} = x^{} \cdot1 = x^{}$

Next, let's simplify the sixth expression $x^{-5} \cdot x^{10}$ This expression can be simplified by applying the base rule.
$x^{-5} \cdot x^{10} = x^{-5+10} = x^{5}$

Now, let's simplify the seventh expression $x^{11} \cdot x^{5}$ This expression can be simplified by applying the base rule.
$x^{11} \cdot x^{5} = x^{11+5} = x^{16}$

Next, let's simplify the eighth expression $\left(x^{3}\right)^{7}$ This expression can be simplified by applying the exponent rule.
$\left(x^{3}\right)^{7} = x^{3 \cdot7} = x^{21}$

Now, let's simplify the ninth expression $x^{-6} \cdot x^{12}$ This expression can be simplified by applying the base rule.
$x^{-6} \cdot x^{12} = x^{-6+12} = x^{6}$

Next, let's simplify the tenth expression $\left(x^{-5}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(x^{-5}\right)^{3} = x^{-5 \cdot3} = x^{-15}$

Now, let's simplify the eleventh expression $\left(x^{11}\right)^{5}$ This expression can be simplified by applying the exponent rule.
$\left(x^{11}\right)^{5} = x^{11 \cdot5} = x^{55}$

Next, let's simplify the twelfth expression $\frac{x^{}}{x^{7}}$
This expression can be simplified by applying the base rule.
$\frac{x^{}}{x^{7}} = x^{-7} = x^{7}$

Now, let's simplify the thirteenth expression $\left(x^{-6}\right)^{4}$ This expression can be simplified by applying the exponent rule.
$\left(x^{-6}\right)^{4} = x^{-6 \cdot4} = x^{-24}$

Next, let's simplify the fourteenth expression $\frac{x^{14}}{x^{-7}}$
This expression can be simplified by applying the base rule.
$\frac{x^{14}}{x^{-7}} = x^{14-(-7)} = x^{21}$

Now, let's simplify the fifteenth expression $\frac{x^{30}}{x^{10}}$
This expression can be simplified by applying the base rule.
$\frac{x^{30}}{x^{10}} = x^{30-10} = x^{20}$

Next, let's simplify the sixteenth expression $\left( x^{3}\right)^{2}$ This expression can be simplified by applying the exponent rule.
$\left( x^{3}\right)^{2} =^{2} \cdot x^{3 \cdot2} =64x^{6}$

Now, let's simplify the seventeenth expression $\frac{x^{30}}{x^{-10}}$
This expression can be simplified by applying the base rule.
$\frac{x^{30}}{x^{-10}} = x^{30-(-10)} = x^{40}$

Next, let's simplify the eighteenth expression $\left(-\frac{4}{x}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(-\frac{4}{x}\right)^{3} = -64x^{-3}$

Now, let's simplify the nineteenth expression $\left(6 x^{4}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(6 x^{4}\right)^{} =36x^{8}$

Next, let's simplify the twentieth expression $\left(-3 x^{} y^{5}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(-3 x^{} y^{5}\right)^{} =9x^{4}y^{10}$

Now, let's simplify the twenty-first expression $\left(-\frac{6}{y}\right)^{}$ This expression can be simplified by applying the exponent rule.
$\left(-\frac{6}{y}\right)^{} = -216y^{-}$

Next, let's simplify the twenty-second expression $\left(3 x^{}\right)\left( x^{7}\right)$ This expression can be simplified by applying the base rule.
$\left(3 x^{}\right)\left( x^{7}\right) =6x^{11}$

Now, let's simplify the twenty-third expression $\left(-3 x^{4} y^{6}\right)^{3}$ This expression can be simplified by applying the exponent rule.
$\left(-3 x^{4} y^{6}\right)^{3} = -27x^{12}y^{18}$

Now, let's simplify the twenty-fifth expression $\left(11 x^{5}\right)\left(9 x^{12}\right)$ This expression can be simplified by applying the base rule.
$\left(11 x^{5}\right)\left(9 x^{12}\right) =99x^{17}$

Next, let's simplify the twenty-sixth expression $\frac{ x^{20}}{ x^{4}}$
This expression can be simplified by applying the base rule.
$\frac{ x^{20}}{ x^{4}} =4x^{16}$

Next, let's simplify the twenty-eighth expression $\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}}$
This expression can be simplified by applying the base rule.
$\frac{25 a^{13} b^{4}}{-5 a^{2} b^{}} = -5a^{11}b$

Now, let's simplify the twenty-ninth expression $\frac{20 x^{24}}{10 x^{6}}$
This expression can be simplified by applying the base rule.
$\frac{20 x^{24}}{10 x^{6}} =2x^{18}$