Mathematics
Grade 11
15 min
Roots of integers
Roots of integers
What you'll learn
- Solve addition and subtraction problems within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction with 80% accuracy.
- Identify and explain patterns in multiplication facts (e.g., multiples of 5 always end in 0 or 5) with at least three different examples.
- Apply multiplication and division within 100 to solve word problems in a group setting, and accurately explain the strategy used to arrive at the solution.
- Recall multiplication facts from 0x0 to 10x10 and division facts related to these multiplication facts, using flashcards, and achieve a score of at least 70% correct within a 2-minute time limit.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define and identify the index, radicand, and principal root of a radical expression.
Evaluate the nth root of any integer, including perfect squares, cubes, and higher powers.
Simplify radical expressions by finding the largest perfect nth power factor of the radicand.
Distinguish between real and imaginary roots for even and odd indices.
Calculate the roots of negative integers, expressing answers using the imaginary unit 'i' when necessary.
Apply the product and quotient rules to simplify expressions involving roots of integers.
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Key Concepts & Vocabulary
TermDefinitionExample
Radical ExpressionAn expression containing a radical symbol (√). It consists of an index, a radicand, and the radical symbol itself.In the expression `∛27`, the entire expression is the radical. `3` is the index, and `27` is the radicand.
IndexThe small number written to the left of the radical symbol that indicates which root is to be taken.In `⁵√32`, the index is 5. If no index is written, it is assumed to be 2 (a square root).
RadicandThe number or expression inside the radical symbol.In `√81`, the radicand is 81. In `∛-64`, the radicand is -64.
Principal RootThe unique, non-negative real root of a non-negative real number. For even indices, the radical symbol denotes only the principal root.The principal square root of 25, written as `√25`, is 5. Although (-5)² =...
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Core Formulas
Definition of nth Root
`ⁿ√a = b` if and only if `bⁿ = a`
This is the fundamental definition connecting radicals and exponents. The nth root of 'a' is the number 'b' that, when raised to the nth power, equals 'a'.
Product Rule for Radicals
`ⁿ√ab = ⁿ√a * ⁿ√b`
Use this rule to split a radical of a product into a product of radicals. This is key for simplifying radicals by factoring out perfect nth powers. (Note: `a` and `b` must be non-negative if `n` is even).
Quotient Rule for Radicals
`ⁿ√(a/b) = (ⁿ√a) / (ⁿ√b)`
Use this rule to split a radical of a quotient into a quotient of radicals. (Note: `a` must be non-negative if `n` is even, and `b` must be positive if `n` is even).
Roots of Negative Numbers
If `n` is odd, `ⁿ√-a = -ⁿ√a`. If...
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Sign Up Free to ContinueSample Practice Questions
Challenging
Simplify the expression `√-12 * √-3`.
A.-6
B.6
C.6i
D.-6i
Challenging
Simplify the expression `∜(1250/2)`.
A.25
B.5√5
C.5
D.5∜2
Challenging
The simplified form of an expression `∛x` is `-4∛2`. What is the integer value of `x`?
A.-64
B.-128
C.-32
D.-256
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Frequently asked questions
What grade level is "Roots of integers"?
Roots of integers is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Roots of integers?
You'll be able to: Solve addition and subtraction problems within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction with 80% accuracy; Identify and explain patterns in….
Is "Roots of integers" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Roots of integers?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.