A comprehensive summary of the topics of Equations and Exponents for Grade 11 Mathematics. Written for easy studying of the theory involved in these topics, with examples. Easy to use and excellent to use for revision purposes.
Real Numbers → ℝ Any number that can be represented on a
number line
Table of Contents Rational → ℚ – Any number that can be written as a
fraction. It stops or recurs (pattern) e.g., 0,6666˙; ⅓
1) The Number System Integers → ℤ – All the whole numbers and their
opposites on the number line, including zero
2) Exponents
e.g: −2 ; +5;
• Laws
Whole Numbers → – Counting numbers, including
• Simplification of exponents separated by factors 0 e.g: 0; 1; 2; 3….
•
Natural → – Counting numbers starting at 1
Simplification of exponents separated by terms
• Exponential Equations e.g 1; 2; 3; 4…….
• Surds Irrational → ℚ1 - A number that does not stop or recur
and and can’t be represented as a fraction
e.g: 0,15896 (No pattern)
3) Equations and Inequalities
• Linear equations
• Quadratic equations
• Equations with fractions Non-Real Numbers
𝑨𝒏𝒚 𝑵𝒖𝒎𝒃𝒆𝒓 𝟑
• K-method Undefined - →
𝟎 𝟎
• Equations with surds Non – Real - √𝑎𝑛𝑦 (− )𝑛𝑢𝑚𝑏𝑒𝑟 → √− 5
• Completing the square
• Quadratic formula
• Simultaneous equations
• Inequalities
• Nature of roots
1| GRADE 11 Claire Bevolo 2021
, EXPONENTS Law 4 → When raising a base to a negative exponent, the base inverts
and the exponent becomes positive
𝟏
𝒙−𝟐 = 𝒙𝟐
Exponential Laws 4
E.g.,1 4𝑥 −3 = → only invert the base with the negative exponent
𝑥3
When applying the laws of exponents, it is important to note that the 5 −4
E.g.,2 = 5𝑥
exponent ONLY belongs to the number that it is attached to →in the 𝑥4
expression 6𝑥 2 only 𝒙 has an exponent of 2 and NOT 6.
Remember to keep the BASES THE SAME! Law 5 → any base raised to the power of zero is equal to 1
𝑥0 = 1
Law 1 → When multiplying numbers with the same base, add the E.g.,1 (6𝑥𝑦 5 )0 = 1
exponents.
Law 6 → when taking the root of a base, divide the exponents
𝑥 2 𝑥 3 = 𝑥 2+3 = 𝑥 5 3
6
√𝑥 6 = 𝑥 3 = 𝑥 2
E.g., 5𝑥 4 × −3𝑥 3 → First do the Signs, then the Numbers and finally
E.g.1 √16𝑥 4 𝑦 6 = 4𝑥 2 𝑦 3 → don’t forget SNV
the Variables (SNV)
Exponents separated by FACTORS
Answer = -15𝑥12
52𝑥−1 9𝑥−2
Law 2 → When dividing numbers with the same base, subtract the
exponents. 152𝑥−3
𝑥 10
= 𝑥 10−2 = 𝑥 8
𝑥2
−6𝑥 5 Step 1 → Prime factorize the bases (Each base must be a prime number)
E.g. = +2𝑥 5−2 = 2𝑥 3
−3𝑥 2 Use your calculator. Enter the number, press =, then press SHIFT and
the °, ,, button.
Law 3 → When raising exponents to another exponent, multiply the
exponents. 52𝑥−1 32(𝑥−2)
( 𝑥 3 )4 = 𝑥 3×4 = 𝑥12 52𝑥−3 32𝑥−3
E.g.1 (4𝑥 3 )2 = 42 𝑥 3×2 = 16𝑥 6 The exponent must go to each base of the prime factors
𝑥2 𝑥 2×2 𝑥4
E.g.2 (𝑦 3)2 = 𝑦 3×2 = 𝑦 6
2| GRADE 11 Claire Bevolo 2021
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