### Chapter 6 Algebraic Expressions and Identities – Exercise 6.3 | Set 1

**Explain each of the products as monomials and verify the result in each case for x = 1**

**Question 18:** (3x) * (4x) * (-5x)

**Solution: **

First, separate the numbers and variables.

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

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

= (-60) * (x

^{1+1+1})= -60x

^{3}

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

Putting x = 1 in LHS we get,

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

= 3 * 4 * -5

= -60

RHS = -60x

^{3}Putting x = 1 in RHS we get,

= -60 * (1)

^{3}

^{= }-60LHS = RHS

Hence, verified.

**Question 19:** (4x^{2}) * (-3x) * ((4/5)x^{3})

**Solution:**

First separate the numbers and variables.

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

^{2}* x * x^{3})Add the powers of the same variable and multiply the numbers.

= (-48/5) * (x

^{2+1+3})= (-48/5)x

^{6}

Verification:LHS = (4x

^{2}) * (-3x) * ((4/5)x^{3})Putting x = 1 in LHS we get,

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

= 4 * -3 * (4/5)

= (-48/5)

RHS = (-48/5)x

^{6}Putting x = 1 in RHS we get,

= (-48/5) * (1)

^{6}= -(48/5)

LHS = RHS

Hence, verified.

**Question 20: **(5x^{4}) * (x^{2 })^{3 }* (2x)^{2}

**Solution:**

First separate the numbers and variables.

= (5 * 4) * (x

^{4}* x^{6}* x^{2})Add the powers of the same variable and multiply the numbers.

= (20) * (x

^{4+6+2})= (20)x

^{12}

Verification:LHS = (5x

^{4}) * (x^{2})^{3}* (2x)^{2}Putting x = 1 in LHS we get,

= (5 * (1)

^{4}) * ((1^{2}))^{3}* (2 * 1)^{2}= (5 * 1) * (1)

^{3}* (2)^{2}= 5 * 1 * 4

= 20

RHS = (20)x

^{12}Putting x = 1 in RHS we get,

= (20) * (1)

^{12}= 20

LHS = RHS

Hence, verified.

**Question 21: **(x^{2 })^{3 }* (2x) * (-4x) * (5)

**Solution:**

First separate the numbers and variables.

= (2 *-4 * 5) * (x

^{6}* x * x)Add the powers of the same variable and multiply the numbers.

= (-40) * (x

^{6+1+1})= (-40)x

^{8}

Verification:LHS = (x

^{2})^{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)x

^{8}Putting x = 1 in RHS we get,

= (-40) * (1)

^{8}= -40

LHS = RHS

Hence, verified.

**Question 22: Write down the product of -8x**^{2}y^{6 }and -20xy. Verify the product for x = 2.5, y = 1.

^{2}y

^{6 }and -20xy. Verify the product for x = 2.5, y = 1.

**Solution: **

(-8x

^{2}y^{6 }) * (-20xy)First separate the numbers and variables.

= (-8 * -20) * (x

^{2}* x) * (y^{6 }* y)Add the powers of the same variable and multiply the numbers.

= 160 * (x

^{2+1}) * (y^{6+1})= 160x

^{3}y^{7}

Verification:LHS = (-8x

^{2}y^{6}) * (-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 = 160x

^{3}y^{7}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.

**Question 23: Evaluate (3.2x**^{6}y^{3}) * (2.1x^{2}y^{2}) when x = 1 and y = 0.5.

^{6}y

^{3}) * (2.1x

^{2}y

^{2}) when x = 1 and y = 0.5.

**Solution: **

First, separate the numbers and variables.

= (3.2 * 2.1) * (x

^{6}* x^{2}) * (y^{3}* y^{2})Add the powers of the same variable and multiply the numbers.

= 6.72 * (x

^{6+2}) * (y^{3+2})= 6.72x

^{8}y^{5}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

**Question 24: Find the value of (5x**^{6}) * (-1.5x^{2}y^{3}) * (-12xy^{2}) when x = 1, y = 0.5.

^{6}) * (-1.5x

^{2}y

^{3}) * (-12xy

^{2}) when x = 1, y = 0.5.

**Solution: **

First, separate the numbers and variables.

= (5 * -1.5 * -12) * (x

^{6}* x^{2 }* x) * (y^{3}* y^{2})Add the powers of the same variable and multiply the numbers.

= 90 * (x

^{6+2+1}) * (y^{3+2})= 90x

^{9}y^{5}Putting x = 1 and y = 0.5 in the result we get

= 90 * (1)

^{9}* (0.5)^{5}= 90 * 1 * 0.03125

= 2.8125

**Question 25: Evaluate when (2.3a**^{5}b^{2}) * ((1.2)a^{2}b^{2}) when a = 1 and b = 0.5.

^{5}b

^{2}) * ((1.2)a

^{2}b

^{2}) when a = 1 and b = 0.5.

**Solution: **

First, separate the numbers and variables.

= (2.3 * 1.2) * (a

^{5 }* a^{2}) * (b^{2}* b^{2})Add the powers of the same variable and multiply the numbers.

= 2.76 * (a

^{5+2}) * (b^{2+2})= 2.76a

^{7}b^{4}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

**Question 26: Evaluate for (-8x**^{2}y^{6}) * (-20xy) when x = 2.5 and y = 1.

^{2}y

^{6}) * (-20xy) when x = 2.5 and y = 1.

**Solution:**

First, separate the numbers and variables.

= (-8 * -20) * (x

^{2}* x) * (y^{6 }* y)Add the powers of the same variable and multiply the numbers.

= 160 * (x

^{2+1}) * (y^{6+1})= 160x

^{3}y^{7}Putting x = 2.5 and y = 1 in the result we get

= 160 * (2.5)

^{3}* (1)^{7}= 160 * 15.625 * 1

= 2500

**Express each of the following products as **monomials** and verify the result for x = 1, y = 2: (27 – 31)**

**Question 27: (-xy ^{3}) * (yx^{3 }) * (xy)**

**Solution:**

First separate the numbers and variables.

= (-1 * 1 * 1) * (x * x

^{3}* x) * (y^{3 }* y * y)Add the powers of the same variable and multiply the numbers.

= -1 * (x

^{1+3+1 }) * (y^{3+1+1})= -x

^{5}y^{5}

Verification:LHS = (-xy

^{3}) * (yx^{3}) * (xy)Putting x = 1 and y = 2 in LHS we get,

= (-1 * (2)

^{3}) * (2 * (1)^{3 }) * (1 * 2)= -8 * 2 * 2

= -32

RHS = -x

^{5}y^{5}Putting x = 1 and y = 2 in RHS we get,

= -1 * (1)

^{5}* (2)^{5}= -32

LHS = RHS

Hence, verified.

**Question 28:** ((1/8) x^{2}y^{4}) * ((1/4) x^{4}y^{2 }) * (xy) * (5)

**Solution:**

First, separate the numbers and variables.

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

^{2}* x^{4}* x) * (y^{4}* y^{2}* y)Add the powers of the same variable and multiply the numbers.

= (5/32) * (x

^{2+4+1}) * (y^{4+2+1})= (5/32)x

^{7}y^{7}

Verification:LHS = ((1/8) x

^{2}y^{4}) * ((1/4) x^{4}y^{2}) * (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)x

^{7}y^{7}Putting x = 1 and y = 2 in RHS we get,

= (5/32) * (1)

^{7 }* (2)^{7}= (5/32) * (128)

= 20

LHS = RHS

Hence, verified

**Question 29: **(2/5)a^{2}b * (-15b^{2}ac) * ((-1/2)c^{2})

**Solution: **

First, separate the numbers and variables.

= ((2/5) * (-15) * (-1/2)) * (a

^{2}* a) * (b* b^{2}) * (c * c^{2})Add the powers of the same variable and multiply the numbers.

= 3 * (a

^{2+1}) * (b^{1+2 }) * (c^{1+2})= 3a

^{3}b^{3}c^{3}This expression does not contain x and y . Hence the result cannot be verified for x = 1 and y = 2.

**Question 30: ((-4/7)a**^{2}b) * ((-2/3)b^{2}c) * ((-7/6)c^{2}a)

^{2}b) * ((-2/3)b

^{2}c) * ((-7/6)c

^{2}a)

**Solution: **

First separate the numbers and variables.

= ((-4/7) * (-2/3) * (-7/6)) * (a

^{2}* a) * (b* b^{2}) * (c * c^{2})Add the powers of the same variable and multiply the numbers.

= (-4/9) * (a

^{2+1}) * (b^{1+2}) * (c^{1+2})= (-4/9)a

^{3}b^{3}c^{3}This expression does not contain x and y . Hence the result cannot be verified for x = 1 and y = 2.

**Question 31:** ((4/9)abc^{3}) * ((-27/5)a^{3}b^{2}) * (-8b^{3}c)

**Solution:**

First, separate the numbers and variables.

= ((4/9) * (-27/5) * (-8)) * (a * a

^{3}) * (b * b^{2 }* b^{3}) * (c^{3}* c)Add the powers of the same variable and multiply the numbers.

= (96/5) * (a

^{1+3}) * (b^{1+2+3}) * (c^{3+1})= (96/5)a

^{4}b^{6}c^{4}This expression does not contain x and y. Hence, the result cannot be verified for x = 1 and y = 2.

**Evaluate each of the following when x = 2 and y = -1.**

**Question 32: **(2xy) * ((x^{2}y) /4) * (x^{2}) * (y^{2})

**Solution: **

First, separate the numbers and variables.

= (2 * (1/4)) * (x * x

^{2 }* x^{2}) * (y * y * y^{2})Add the powers of the same variable and multiply the numbers.

= (1/2) * (x

^{1+2+2}) * (y^{1+1+2})= (1/2)x

^{5}y^{4}Putting x = 2 and y = -1 in the result we get,

= (1/2) * ( 2)

^{5 }* (-1)^{4}= 16

**Question 33: **(3/5)x^{2}y * ((-15/4) * x * y^{2}) * ((7/9) x^{2}y^{2})

**Solution: **

First, separate the numbers and variables.

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

^{2}* x * x^{2}) * (y * y^{2}* y^{2})Add the powers of the same variable and multiply the numbers.

= (-7/4) * (x

^{2+1+2}) * (y^{1+2+2})= (-7/4)x

^{5}y^{5}Putting x = 2 and y = -1 in the result we get,

= (-7/4) * ( 2)

^{5}* (-1)^{5}= -56