QuestionSimplify the following expressions with only positive exponents:
1) $2 m^{2} \cdot 2 m^{3}$ 2) $m^{4} \cdot 2 m^{-3}$ 3) $4 r^{-3} \cdot 2 r^{2}$ 4) $4 n^{4} \cdot 2 n^{-3}$ 5) $2 k^{4}, 4 k$ 6) $2 x^{3} y^{-3} \cdot 2 x^{-1} y^{3}$ 7) $2 y^{2}+3 x$ 8) $4 v^{3} \cdot v u^{2}$ 9) $4 a^{3} b^{2}+3 a^{-4} b^{-3}$ 10) $x^{2} y^{-4} \cdot x^{3} y^{2}$

Studdy Solution

STEP 1

Assumptions1. We are using the properties of exponents, which state that when multiplying terms with the same base, we add the exponents. . We are assuming that all variables represent real numbers.
3. We are assuming that none of the variables in the denominators are equal to zero.

STEP 2

implify the first expression $2 m^{2} \cdot2 m^{}$ .
According to the properties of exponents, we add the exponents of $m$ and multiply the constants.
$2 m^{2} \cdot2 m^{} =4 m^{2+}$

STEP 3

Calculate the exponent.
$m^{2+3} = m^{5}$

STEP 4

implify the second expression $m^{4} \cdot2 m^{-3}$ .
Again, we add the exponents of $m$ .
$m^{4} \cdot2 m^{-3} =2 m^{4-3}$

STEP 5

Calculate the exponent.
$2 m^{4-3} =2 m^{1} =2m$

STEP 6

implify the third expression $4 r^{-3} \cdot2 r^{2}$ .
We add the exponents of $r$ and multiply the constants.
$4 r^{-3} \cdot2 r^{2} =8 r^{-3+2}$

STEP 7

Calculate the exponent.
$r^{-3+2} = r^{-1}$

STEP 8

Since the answer should contain only positive exponents, we rewrite $r^{-1}$ as $\frac{1}{r}$ .
$8 r^{-1} = \frac{8}{r}$

STEP 9

implify the fourth expression $4 n^{4} \cdot2 n^{-3}$ .
We add the exponents of $n$ and multiply the constants.
$4 n^{4} \cdot2 n^{-3} =8 n^{4-3}$

STEP 10

Calculate the exponent.
$8 n^{4-3} =8 n^{} =8n$

STEP 11

implify the fifth expression $k^{4},4 k$ .
This expression is already simplified as it is. There is no operation between the two terms.

STEP 12

implify the sixth expression $2 x^{} y^{-} \cdot2 x^{-} y^{}$ .
We add the exponents of $x$ and $y$ separately and multiply the constants.
$2 x^{} y^{-} \cdot2 x^{-} y^{} =4 x^{-} y^{-+}$

STEP 13

Calculate the exponents.
$x^{3-} y^{-3+3} = x^{2} y^{0}$

STEP 14

Since any number (except zero) to the power of zero is, we simplify $y^{0}$ to.
$4 x^{2} y^{0} =4 x^{2} \cdot =4 x^{2}$

STEP 15

implify the seventh expression $2 y^{2}+3 x$ .
This expression is already simplified as it is. There is no operation between the two terms.

STEP 16

implify the eighth expression $4 v^{3} \cdot v u^{2}$ .
We multiply the constants and the terms with the same base.
$4 v^{3} \cdot v u^{2} =4 v^{3+} u^{2}$

STEP 17

Calculate the exponent.
$4 v^{3+} u^{2} =4 v^{4} u^{2}$

STEP 18

implify the ninth expression $4 a^{3} b^{2}+3 a^{-4} b^{-3}$ .
This expression is already simplified as it is. There is no operation between the two terms.

STEP 19

implify the tenth expression $x^{} y^{-4} \cdot x^{3} y^{}$ .
We add the exponents of $x$ and $y$ separately.
$x^{} y^{-4} \cdot x^{3} y^{} = x^{+3} y^{-4+}$

STEP 20

Calculate the exponents.
$x^{+3} y^{-4+} = x^{5} y^{-}$

STEP 21

Since the answer should contain only positive exponents, we rewrite $y^{-}$ as $\frac{1}{y^{}}$ .
$x^{5} y^{-} = x^{5} \cdot \frac{1}{y^{}} = \frac{x^{5}}{y^{}}$ The simplified expressions are1) $4 m^{5}$ ) $m$ 3) $\frac{8}{r}$ 4) $8n$ 5) $k^{4},4 k$ 6) $4 x^{}$ 7) $y^{}+3 x$ 8) $4 v^{4} u^{}$ 9) $4 a^{3} b^{}+3 a^{-4} b^{-3}$ 10) $\frac{x^{5}}{y^{}}$

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Studdy Solution

STEP 1

Assumptions1. We are using the properties of exponents, which state that when multiplying terms with the same base, we add the exponents. . We are assuming that all variables represent real numbers.3. We are assuming that none of the variables in the denominators are equal to zero.

STEP 2

implify the first expression 2m2⋅2m2 m^{2} \cdot2 m^{}2m2⋅2m.According to the properties of exponents, we add the exponents of mmm and multiply the constants.2m2⋅2m=4m2+2 m^{2} \cdot2 m^{} =4 m^{2+}2m2⋅2m=4m2+

STEP 3

Calculate the exponent.m2+3=m5 m^{2+3} = m^{5}m2+3=m5

STEP 4

implify the second expression m4⋅2m−3m^{4} \cdot2 m^{-3}m4⋅2m−3.Again, we add the exponents of mmm.m4⋅2m−3=2m4−3m^{4} \cdot2 m^{-3} =2 m^{4-3}m4⋅2m−3=2m4−3

STEP 5

Calculate the exponent.2m4−3=2m1=2m2 m^{4-3} =2 m^{1} =2m2m4−3=2m1=2m

STEP 6

implify the third expression 4r−3⋅2r24 r^{-3} \cdot2 r^{2}4r−3⋅2r2.We add the exponents of rrr and multiply the constants.4r−3⋅2r2=8r−3+24 r^{-3} \cdot2 r^{2} =8 r^{-3+2}4r−3⋅2r2=8r−3+2

STEP 7

Calculate the exponent.r−3+2=r−1 r^{-3+2} = r^{-1}r−3+2=r−1

STEP 8

Since the answer should contain only positive exponents, we rewrite r−1r^{-1}r−1 as 1r\frac{1}{r}r1​.8r−1=8r8 r^{-1} = \frac{8}{r}8r−1=r8​

STEP 9

implify the fourth expression 4n4⋅2n−34 n^{4} \cdot2 n^{-3}4n4⋅2n−3.We add the exponents of nnn and multiply the constants.4n4⋅2n−3=8n4−34 n^{4} \cdot2 n^{-3} =8 n^{4-3}4n4⋅2n−3=8n4−3

STEP 10

Calculate the exponent.8n4−3=8n=8n8 n^{4-3} =8 n^{} =8n8n4−3=8n=8n

STEP 11

implify the fifth expression k4,4k k^{4},4 kk4,4k.This expression is already simplified as it is. There is no operation between the two terms.

STEP 12

implify the sixth expression 2xy−⋅2x−y2 x^{} y^{-} \cdot2 x^{-} y^{}2xy−⋅2x−y.We add the exponents of xxx and yyy separately and multiply the constants.2xy−⋅2x−y=4x−y−+2 x^{} y^{-} \cdot2 x^{-} y^{} =4 x^{-} y^{-+}2xy−⋅2x−y=4x−y−+

STEP 13

Calculate the exponents.x3−y−3+3=x2y0 x^{3-} y^{-3+3} = x^{2} y^{0}x3−y−3+3=x2y0

STEP 14

Since any number (except zero) to the power of zero is, we simplify y0y^{0}y0 to.4x2y0=4x2⋅=4x24 x^{2} y^{0} =4 x^{2} \cdot =4 x^{2}4x2y0=4x2⋅=4x2

STEP 15

implify the seventh expression 2y2+3x2 y^{2}+3 x2y2+3x.This expression is already simplified as it is. There is no operation between the two terms.

STEP 16

implify the eighth expression 4v3⋅vu24 v^{3} \cdot v u^{2}4v3⋅vu2.We multiply the constants and the terms with the same base.4v3⋅vu2=4v3+u24 v^{3} \cdot v u^{2} =4 v^{3+} u^{2}4v3⋅vu2=4v3+u2

STEP 17

Calculate the exponent.4v3+u2=4v4u24 v^{3+} u^{2} =4 v^{4} u^{2}4v3+u2=4v4u2

STEP 18

implify the ninth expression 4a3b2+3a−4b−34 a^{3} b^{2}+3 a^{-4} b^{-3}4a3b2+3a−4b−3.This expression is already simplified as it is. There is no operation between the two terms.

STEP 19

implify the tenth expression xy−4⋅x3yx^{} y^{-4} \cdot x^{3} y^{}xy−4⋅x3y.We add the exponents of xxx and yyy separately.xy−4⋅x3y=x+3y−4+x^{} y^{-4} \cdot x^{3} y^{} = x^{+3} y^{-4+}xy−4⋅x3y=x+3y−4+

STEP 20

Calculate the exponents.x+3y−4+=x5y−x^{+3} y^{-4+} = x^{5} y^{-}x+3y−4+=x5y−

STEP 21

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Assumptions1. We are using the properties of exponents, which state that when multiplying terms with the same base, we add the exponents. . We are assuming that all variables represent real numbers.
3. We are assuming that none of the variables in the denominators are equal to zero.

implify the first expression $2 m^{2} \cdot2 m^{}$ .
According to the properties of exponents, we add the exponents of $m$ and multiply the constants.
$2 m^{2} \cdot2 m^{} =4 m^{2+}$

Calculate the exponent.
$m^{2+3} = m^{5}$

implify the second expression $m^{4} \cdot2 m^{-3}$ .
Again, we add the exponents of $m$ .
$m^{4} \cdot2 m^{-3} =2 m^{4-3}$

Calculate the exponent.
$2 m^{4-3} =2 m^{1} =2m$

implify the third expression $4 r^{-3} \cdot2 r^{2}$ .
We add the exponents of $r$ and multiply the constants.
$4 r^{-3} \cdot2 r^{2} =8 r^{-3+2}$

Calculate the exponent.
$r^{-3+2} = r^{-1}$

Since the answer should contain only positive exponents, we rewrite $r^{-1}$ as $\frac{1}{r}$ .
$8 r^{-1} = \frac{8}{r}$

implify the fourth expression $4 n^{4} \cdot2 n^{-3}$ .
We add the exponents of $n$ and multiply the constants.
$4 n^{4} \cdot2 n^{-3} =8 n^{4-3}$

Calculate the exponent.
$8 n^{4-3} =8 n^{} =8n$

implify the fifth expression $k^{4},4 k$ .
This expression is already simplified as it is. There is no operation between the two terms.

implify the sixth expression $2 x^{} y^{-} \cdot2 x^{-} y^{}$ .
We add the exponents of $x$ and $y$ separately and multiply the constants.
$2 x^{} y^{-} \cdot2 x^{-} y^{} =4 x^{-} y^{-+}$

Calculate the exponents.
$x^{3-} y^{-3+3} = x^{2} y^{0}$

Since any number (except zero) to the power of zero is, we simplify $y^{0}$ to.
$4 x^{2} y^{0} =4 x^{2} \cdot =4 x^{2}$

implify the seventh expression $2 y^{2}+3 x$ .
This expression is already simplified as it is. There is no operation between the two terms.

implify the eighth expression $4 v^{3} \cdot v u^{2}$ .
We multiply the constants and the terms with the same base.
$4 v^{3} \cdot v u^{2} =4 v^{3+} u^{2}$

Calculate the exponent.
$4 v^{3+} u^{2} =4 v^{4} u^{2}$

implify the ninth expression $4 a^{3} b^{2}+3 a^{-4} b^{-3}$ .
This expression is already simplified as it is. There is no operation between the two terms.

implify the tenth expression $x^{} y^{-4} \cdot x^{3} y^{}$ .
We add the exponents of $x$ and $y$ separately.
$x^{} y^{-4} \cdot x^{3} y^{} = x^{+3} y^{-4+}$

Calculate the exponents.
$x^{+3} y^{-4+} = x^{5} y^{-}$