Objective:
Language and tools of algebra is the first chapter in Heart of algebra section. The student should be able to solve questions related to the following topics after learning this section.
1 Variables and Expressions
2 Simplifying Algebraic Expressions
3 Rational, Irrational, and Decimal
4 Exponents and Order of Operations
Alphabets and symbols are variables used to represent numbers. Let us review writing algebraic expressions and equations. Please check Basic algebra in pre-algebra to get worksheets on this topic
- One-fourth of square of x
Square of x = x2
One-fourth of square of x =
\frac{1}4x^22. Two-fifth of cube of a number x added to 5 times the number x
Cube of x = x3
5 times the number x = 5x
Two-fifth of cube of number x =
\:\frac{2}{5}\:x^3Two-fifth of cube of a number x added to 5 times the number x
\:\frac{2}{5}\:x^3+5x- If length of a rectangle is twice its width w. If the perimeter of the rectangle is P, what is the perimeter of the rectangle in terms of w
Width of rectangle = w
Length of rectangle = twice the width = 2w
Perimeter of rectangle = 2(length)+ 2(width)
P = 2(2w) + 2(w)
= 4w +2w
P = 6w
Combining Like terms
Combining like terms means combine the terms with same variable part.
3xy and -5xy has variable part xy. So, combining them 3xy + (-5xy) = -2xy
- Simplify
-3x^3 y+4xy^2-3xy+2x^3 y-4xy
=(-3 x^3 y+2x^3 y)+4xy^2-3xy-4xy
= -x^3 y+4xy^2-7xy
2. Simplify
4ab+3a^2b-2ab+2a^2b+4
=3a^2b+2a^2b+4ab-2ab+4
=5a^2b+2ab+4
3. Combine like terms
2x^3-4x^2+3x-5-5x^3+2x^2-7x-4+6x
=-3x^3-2x^2+2x-9
4. Simplify
3x^2 y^2+3y^2 x+2x^2 y^2-3xy^2-y^2 x^2+y^3 x
=3x^2y^2+2x^2y^2-x^2y^2+xy^3+3xy^2-3xy^2
=4x^2y^2+xy^3+3xy^2-3xy^2
=4x^2y^2+xy^3
Order of Operations
Step 1: Evaluate and combine all the expressions within the Parentheses and other brackets first;
Step 2: Simplify and assess all expressions involving Exponents;
Step 3: Do all the remaining Multiplication & Division, as one comes to them, when working from left to right in the expressions;
Step 4: Finally, carry out the remaining Addition & Subtraction, as you approach them, when working from left to right in the expression.
The above procedure will result in the final simplified expression or solution for the given equation. As the “ORDER OF OPERATIONS “
4 steps – simplifying the Parentheses, Exponents, (Multiplication & division) and (Addition & Subtraction) symbols in the expression, in the above order, it can be simply remembered as “PEMDAS “.
Number sense (Rational, Irrational and decimal)
Rational numbers are numbers which can be written in the form of Where p and q are integers.
Whole numbers, integers, fractions are part of rational numbers.
Exponents
- Simplify
m^8\left(m^{5\:}n^3\right).\:n^4=m^{13}n^72. Find the value of :
16^{-\frac{3}{4}}\:.\:16^{\frac{1}{4}}\:= 16^{-\frac{3}{4}+\frac{1}{4}}=16^{-\frac{1}{2}}=4^{2\left(-\frac{1}{2}\right)}=4^{-1}=\frac{1}{4}3. Simplify
\left(a^{\frac{2}{3}}b^{\frac{1}{4}}\:\cdot \:\:b^{-\frac{1}{3}}c^{-\frac{1}{2}}\cdot \:a^{\frac{3}{4}}c^{\frac{1}{3}}\right)^{-\frac{3}{4}}=\frac{1}{\left(a^{\frac{2}{3}}b^{\frac{1}{4}}b^{-\frac{1}{3}}c^{-\frac{1}{2}}a^{\frac{3}{4}}c^{\frac{1}{3}}\right)^{\frac{3}{4}}}=\frac{b^{\frac{1}{16}}c^{\frac{1}{8}}}{a^{\frac{17}{16}}}4. Simplify
u^{-4}v^8\left(\frac{8u^{-5}v^2}{4u^4v^{-7}}\right)=\frac{8u^{-5}v^2}{4u^4v^{-7}}\cdot \frac{1}{u^4}v^8=\frac{2v^9}{u^9}\cdot \frac{1}{u^4}v^8=\frac{2v^{17}}{u^{13}}Operations on rational numbers
Simplify
Irrational Numbers
Every number expressed in decimal form are expressed as non-terminating and non-repeating decimals are irrational numbers
Radical numbers (square root of non-negative non-perfect square number) are irrational numbers.
- Radical numbers
\sqrt{2\:},\sqrt{3\:},\sqrt{5\:},\sqrt{7\:},\sqrt{11\:}........2. All non-terminating non-repeating numbers
a) 0.010010001000100001…
b) 0. 21321121113211113..