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Grade 11 - Mathematics Summary on Equations and Exponents
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A comprehensive summary, colorful summary, with examples, covering Equations and Exponents. Brilliant for exam and test revision.
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THE NUMBER SYSTEMReal 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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