8类RD Sharma解决方案–第6章代数表达式和恒等式–练习6.3 |套装2

第6章代数表达式和恒等式–练习6.3 |套装1

以单项式解释每种产品，并在每种情况下针对x = 1验证结果

问题18： (3x)*(4x)*(-5x)

解决方案：

First, separate the numbers and variables.

= (3 * 4 * -5) * (x * x * x)

Add the powers of the same variable and multiply the numbers.

= (-60) * (x¹⁺¹⁺¹)

= -60x³ 

Verification:

LHS = (3x) * (4x) * (-5x)

Putting x = 1 in LHS we get,

= (3 * 1) * (4 * 1) * (-5 * 1)

= 3 * 4 * -5

= -60

RHS = -60x³

Putting x = 1 in RHS we get,

= -60 * (1)³

⁼ -60

LHS = RHS

Hence, verified.

问题19： (4x ² )*(-3x)*((4/5)x ³ )

解决方案：

First separate the numbers and variables.

= (4 * -3 * (4/5)) * (x² * x * x³)

Add the powers of the same variable and multiply the numbers.

= (-48/5) * (x²⁺¹⁺³)

= (-48/5)x⁶

Verification:

LHS = (4x²) * (-3x) * ((4/5)x³)

Putting x = 1 in LHS we get,

= (4 * 1) * (-3 * 1) * ((4/5) * 1)

= 4 * -3 * (4/5)

= (-48/5)

RHS = (-48/5)x⁶

Putting x = 1 in RHS we get,

= (-48/5) * (1)⁶

= -(48/5)

LHS = RHS

Hence, verified.

问题20： (5x ⁴ )*(x ² ) ³ *(2x) ²

解决方案：

First separate the numbers and variables.

= (5 * 4) * (x⁴ * x⁶ * x²)

Add the powers of the same variable and multiply the numbers.

= (20) * (x⁴⁺⁶⁺²)

= (20)x¹²

Verification:

LHS = (5x⁴) * (x²)³  * (2x)²

Putting x = 1 in LHS we get,

= (5 * (1)⁴) * ((1²))³ * (2 * 1)²

= (5 * 1) * (1)³ * (2)²

= 5 * 1 * 4

= 20

RHS = (20)x¹²

Putting x = 1 in RHS we get,

= (20) * (1)¹²

= 20

LHS = RHS

Hence, verified.

问题21： (x ² ) ³ *(2x)*(-4x)*(5)

解决方案：

First separate the numbers and variables.

= (2 *-4  * 5) * (x⁶ * x * x)

Add the powers of the same variable and multiply the numbers.

= (-40) * (x⁶⁺¹⁺¹)

= (-40)x⁸

Verification:

LHS = (x²)³ * (2x) * (-4x) * (5) 

Putting x = 1 in LHS we get,

= (1)⁶ *  (2 * 1) * (-4 * 1) * (5)

= 1 * 2 * -4 * 5

=  -40

RHS = (-40)x⁸

Putting x = 1 in RHS we get,

= (-40) * (1)⁸

= -40

LHS = RHS

Hence, verified.

问题22：记下-8x ² y ⁶和-20xy的乘积。验证乘积x = 2.5，y = 1。

解决方案：

(-8x²y⁶ ) * (-20xy)

First separate the numbers and variables.

= (-8 * -20) * (x² * x) * (y⁶ * y)

Add the powers of the same variable and multiply the numbers.

= 160 * (x²⁺¹) * (y⁶⁺¹)

= 160x³y⁷

Verification:

LHS = (-8x²y⁶) * (-20xy)

Putting x = 2.5 and y = 1 in LHS we get,

= (-8 * (2.5)² * (1)⁶) * (-20 * 2.5 * 1)

= (-8 * 6.25 * 1) * (-20 * 25)

= -50 * -50

= 2500

RHS = 160x³y⁷

Putting x = 2.5 and y = 1 in RHS we get,

= -160 * (2.5)³ * (1)⁷

= -160 * 15.625

= 2500

LHS = RHS

Hence, verified.

问题23：当x = 1且y = 0.5时，^{评估(3.2x 6} y ³ )*(2.1x ² y ^2)。

解决方案：

First, separate the numbers and variables.

= (3.2 * 2.1) * (x⁶ * x²) * (y³ * y²)

Add the powers of the same variable and multiply the numbers.

= 6.72 * (x⁶⁺²) * (y³⁺²)

= 6.72x⁸y⁵

Putting x = 1 and y = 0.5 in the result we get

= 6.72 * (1)⁸ * (0.5)⁵ 

= 6.72 * 0.03125

= 0.21

问题24：当x = 1，y = 0.5时，^{求出(5x 6} )*(-1.5x ² y ³ )*(-12xy ^2)的值。

解决方案：

First, separate the numbers and variables.

= (5 * -1.5 * -12) * (x⁶ * x²  * x) * (y³ * y²)

Add the powers of the same variable and multiply the numbers.

= 90 * (x⁶⁺²⁺¹) * (y³⁺²)

= 90x⁹y⁵

Putting x = 1 and y = 0.5 in the result we get

= 90 * (1)⁹ * (0.5)⁵

= 90 * 1 * 0.03125

= 2.8125

问题25：当a = 1和b = 0.5时^{评估(2.3a 5} b ² )*((1.2)a ² b ^{2)的时间。}

解决方案：

First, separate the numbers and variables.

= (2.3 * 1.2) * (a⁵  * a²) * (b² * b²)

Add the powers of the same variable and multiply the numbers.

= 2.76 * (a⁵⁺²) * (b²⁺²)

= 2.76a⁷b⁴

Putting a = 1 and b = 0.5 in the result we get

= 2.76 * (1)⁷ * (0.5)⁴

= 2.76 * 1 * 0.0625

= 0.1725

问题26：当x = 2.5且y = 1时^{评估(-8x 2} y ^6)*(-20xy)

解决方案：

First, separate the numbers and variables.

= (-8 * -20) * (x²  * x) * (y⁶ * y)

Add the powers of the same variable and multiply the numbers.

= 160 * (x²⁺¹) * (y⁶⁺¹)

= 160x³y⁷

Putting x = 2.5 and y = 1 in the result we get

= 160 * (2.5)³ * (1)⁷

= 160 * 15.625 * 1

= 2500

将以下每个产品表示为单项式，并验证x = 1，y = 2的结果：(27 – 31)

问题27：(-xy ³ )*(yx ³ )*(xy)

解决方案：

First separate the numbers and variables.

= (-1 * 1 * 1) * (x * x³ * x) * (y³ * y * y)

Add the powers of the same variable and multiply the numbers.

= -1 * (x¹⁺³⁺¹ ) * (y³⁺¹⁺¹)

= -x⁵y⁵

Verification:

LHS = (-xy³) * (yx³) * (xy)

Putting x = 1 and y = 2 in LHS we get,

= (-1 * (2)³) * (2 * (1)³ ) * (1 * 2)

= -8 * 2 * 2

= -32

RHS = -x⁵y⁵

Putting x = 1 and y = 2 in RHS we get,

= -1 * (1)⁵ * (2)⁵

= -32

LHS = RHS

Hence, verified.

问题28： ((1/8)x ² y ⁴ )*((1/4)x ⁴ y ² )*(xy)*(5)

解决方案：

First, separate the numbers and variables.

= ((1/8) * (1/4) * 1 * 5) * (x² * x⁴ * x) * (y⁴ * y² * y)

Add the powers of the same variable and multiply the numbers.

= (5/32) * (x²⁺⁴⁺¹) * (y⁴⁺²⁺¹)

= (5/32)x⁷ y⁷

Verification:

LHS = ((1/8) x²y⁴) * ((1/4) x⁴y²) * (xy) * (5)

Putting x = 1 and y = 2 in LHS we get,

= ((1/8) * (1)² * (2)⁴) * ((1/4) * (1)⁴ * (2)²) * (1 * 2) * (5)

= 2 * 1 * 2 * 5

= 20

RHS = (5/32)x⁷y⁷

Putting x = 1 and y = 2 in RHS we get,

= (5/32) * (1)⁷  * (2)⁷

= (5/32) * (128)

= 20

LHS = RHS

Hence, verified

问题29： (2/5)a ² b *(-15b ² ac)*((-1/2)c ² )

解决方案：

First, separate the numbers and variables.

= ((2/5) * (-15) * (-1/2)) * (a² * a) * (b* b²) * (c * c²) 

Add the powers of the same variable and multiply the numbers.

= 3 * (a²⁺¹) * (b¹⁺² ) * (c¹⁺²)

= 3a³b³c³

This expression does not contain  x and y . Hence the result cannot be verified for x = 1 and y = 2.

问题30：((-4/7)a ² b)*((-2/3)b ² c)*((-7/6)c ² a)

解决方案：

First separate the numbers and variables.

= ((-4/7) * (-2/3) * (-7/6)) * (a² * a) * (b* b²) * (c * c²)

Add the powers of the same variable and multiply the numbers.

= (-4/9) * (a²⁺¹) * (b¹⁺²) * (c¹⁺²)

= (-4/9)a³b³c³

This expression does not contain  x and y . Hence the result cannot be verified for x = 1 and y = 2.

问题31： ((4/9)abc ³ )*((–27 / 5)a ³ b ² )*(-8b ³ c)

解决方案：

First, separate the numbers and variables.

= ((4/9) * (-27/5) * (-8)) * (a * a³) * (b * b²  * b³) * (c³ * c)

Add the powers of the same variable and multiply the numbers.

= (96/5) * (a¹⁺³) * (b¹⁺²⁺³) * (c³⁺¹)

= (96/5)a⁴b⁶c⁴

This expression does not contain x and y. Hence, the result cannot be verified for x = 1 and y = 2.

当x = 2和y = -1时，请评估以下各项。

问题32： (2xy)*((x ² y)/ 4)*(x ² )*(y ² )

解决方案：

First, separate the numbers and variables.

= (2 * (1/4)) * (x * x²  * x²) * (y * y * y²)

Add the powers of the same variable and multiply the numbers.

= (1/2) * (x¹⁺²⁺²) * (y¹⁺¹⁺²) 

= (1/2)x⁵y⁴

Putting x = 2 and y = -1 in the result we get,

= (1/2) * ( 2)⁵  * (-1)⁴

= 16

问题33： (3/5)x ² y *((-15/4)* x * y ² )*((7/9)x ² y ² )

解决方案：

First, separate the numbers and variables.

= ((3/5) * (-15/4) * (7/9)) * (x² * x * x²) * (y * y² * y²)

Add the powers of the same variable and multiply the numbers.

= (-7/4) * (x²⁺¹⁺²) * (y¹⁺²⁺²)

= (-7/4)x⁵y⁵

Putting x = 2 and y = -1 in the result we get,

= (-7/4) * ( 2)⁵  * (-1)⁵

= -56