Class 8 RD Sharma Solutions Chapter 6 Algebraic Expressions And Identities Exercise 6.3 | Set 1 (original) (raw)
Last Updated : 11 Sep, 2024
Chapter 6 of RD Sharma's Class 8 mathematics textbook focuses on Algebraic Expressions and Identities. Exercise 6.3 delves into the application of algebraic identities, particularly (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². These identities are fundamental in algebra and form the basis for more complex mathematical operations. Understanding and applying these identities correctly is crucial for solving a wide range of algebraic problems efficiently. By working through the problems in Exercise 6.3, students will develop not only their computational skills but also their analytical thinking and pattern recognition abilities, which are invaluable in mathematical reasoning.
**Find each of the following products:
**Question 1: 5x2 * 4x3
**Solution:
First, separate the numbers and variables.
= (5 * 4) * (x2 * x3)
Add the powers of the same variable and multiply the numbers.
= 20 * (x2+3)
= 20x5
**Question 2: -3a2 * 4b4
**Solution:
First, separate the numbers and variables.
= (-3 * 4) * (a2) * (b4)
= -12a2b4
**Question 3: (-5xy) * (-3x2yz)
**Solution:
First separate the numbers and variables.
= (-5 * -3) * (x * x2) * (y * y) * (z)
Add the powers of the same variable and multiply the numbers.
= 15 * (x1+2) * (y1+1) * (z)
= 15x3y2z
**Question 4: (1/2)xy * (2/3)x2yz2
**Solution:
First separate the numbers and variables.
= ((1/2) * (2/3)) * (x * x2) * (y * y) * (z2)
Add the powers of the same variable and multiply the numbers.
= (1/3) * (x1+2) * (y1+1) * (z2)
= (1/3)x3y2z2
**Question 5: ((-7/5)xy2z) * ((13/3)x2yz2)
**Solution:
First separate the numbers and variables.
= ((-7/5) * (13/3)) * (x * x2) * (y2 * y) * (z * z2)
Add the powers of the same variable and multiply the numbers.
= (-91/15) * (x1+2) * (y2 +1) * (z1+2)
= (-91/15)x3y3z3
**Question 6:(( -24/25 )x3z) * ((-15/16 )xz2y)
**Solution:
First separate the numbers and variables.
= ((-24/25) * (-15/16)) * (x3 * x) * (z * z2) * (y)
Add the powers of the same variable and multiply the numbers.
= (9/4) * (x3+1) * (z1+2) * (y)
= (9/10)x4yz3
**Question 7: ((-1/27)a2b2) * ((9/2))a3b2c2)
**Solution:
First separate the numbers and variables.
= ((-1/27)) * (9/2)) * (a2 * a3) * (b2 * b2) * (c2)
Add the powers of the same variable and multiply the numbers.
= (-1/6) * (a2+3) * (b2+2) * (c2)
= (-1/6)a5b4c2
**Question 8: (-7xy) * ((1/4)x2yz)
**Solution:
First separate the numbers and variables.
= (-7 * (1/4)) * (x * x2) * (y * y) * (z)
Add the powers of the same variable and multiply the numbers.
= (-7 * (1/4)) * (x1+2) * (y1+1) * (z)
= (-7/4) x3y2z
**Question 9: (7ab) * (-5ab2c) * (6abc2)
**Solution:
First separate the numbers and variables.
= (7 * -5 * 6) * (a * a * a) * (b * b2 * b) * (c * c2)
Add the powers of the same variable and multiply the numbers.
= -210 * (a1+1+1) * (b1+2+1) * (c1+2)
= -210a3b4c3
**Question 10: (-5a) * (-10a2) * (-2a3)
**Solution:
First separate the numbers and variables.
= (-5 * -10 * -2) * (a * a2 * a3)
Add the powers of the same variable and multiply the numbers.
= (-100) * (a1+2+3)
= -100a6
**Question 11: (-4x2) * (-6xy2) * (3yz2)
**Solution:
First separate the numbers and variables.
= (-4 * -6 * 3) * (x2 * x) * (y2 * y) * (z2)
Add the powers of the same variable and multiply the numbers.
= -72 * (x2+1) * (y2+1) * (z2)
= -72x3y3z2
**Question 12: ((-2/7 )a4) * ((-3/4 )a2b) * ((-14/5 )b2)
**Solution:
First separate the numbers and variables.
= ((-2/7) * (-3/4) * (-14/5)) * (a4 * a2) * (b * b2)
Add the powers of the same variable and multiply the numbers.
= (-3/5) * (a4+2) * (b1+2)
= (-3/5)a6b3
**Question 13: ((7/9)ab2) * ((15/7) ac2b) * ((-3/5)a2c)
**Solution:
First separate the numbers and variables.
= ((7/9) * (15/7) * (-3/5)) * (a * a * a2) * (b2 * b) * (c2 * c)
Add the powers of the same variable and multiply the numbers.
= (-1) * (a1+1+2) * (b2+1) * (c2+1)
= -a4b3c3
**Question 14: ((4/3)u2vw) * (-5uvw2) * ((1/3)v2wu)
**Solution:
First separate the numbers and variables.
= ((4/3) * (-5) * (1/3)) * (u2 * u * u) * (v * v * v2) * (w * w2 * w)
Add the powers of the same variable and multiply the numbers.
= (-20/9) * (u2+1+1) * (v1+1+2) * (w1+2+1)
= (-20/9)u4v4w4
**Question 15: (0.5x) * ((1/3) xy2z4) * (24x2yz)
**Solution:
First separate the numbers and variables.
= ((0.5) * (1/3) * (24)) * (x * x * x2) * (y2 * y) * (z4 * z)
Add the powers of the same variable and multiply the numbers.
= (4) * (x1+1+2) * (y2+1) * (z4+1)
= 4x4y3z5
**Question 16: ((4/3)pq2) * ((-1/4)p2r) * (16p2q2r2)
**Solution:
First separate the numbers and variables.
= (( 4/3) * (-1/4) * (16)) * (p * p2 * p2) * (q2 * q2) * (r * r2)
Add the powers of the same variable and multiply the numbers.
= (-16/3) * (p1+2+2) * (q2+2) * (r1+2)
= (-16/3)p5q4r3
**Question 17: (2.3xy ) * ( 0.1x ) * ( 0.16 )
**Solution:
First separate the numbers and variables.
= (2.3 * 0.1 * 0.16) * (x * x) * (y)
Add the powers of the same variable and multiply the numbers.
= 0.0368 * (x1+1) * (y)
= 0.0368x2y
Summary
Exercise 6.3 in Chapter 6 of RD Sharma's Class 8 mathematics textbook provides students with a comprehensive set of problems to practice applying algebraic identities. These questions range from simple expansions to more complex applications, helping students develop a strong foundation in algebraic manipulation and problem-solving skills.Exercise 6.3 in Chapter 6 of RD Sharma's Class 8 mathematics textbook serves as a crucial stepping stone in a student's journey through algebra. This exercise focuses on the application and mastery of two fundamental algebraic identities: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². These identities are not mere formulas to be memorized, but powerful tools that unlock efficient problem-solving techniques in algebra and beyond.The problems in this exercise are carefully curated to provide a comprehensive learning experience. They range from straightforward expansions of binomial squares to more complex applications involving fractions, radicals, and decimal numbers. This variety ensures that students not only learn to mechanically apply the identities but also develop the ability to recognize situations where these identities can be useful.