Index Forms with a Specified Base

By the end of this lesson, learners should be able to

1. Factor a quantity toward a specified base.
2. Express powers with bases eight, eleven, or thirteen without copying examples.
3. Verify a specified-base representation.

Key vocabulary

Specified base

The base a problem requires you to use.

A prompt may require 4,096 written with base eight.

Quotient

The result obtained after dividing one number by another.

Dividing 1,331 by eleven repeatedly produces whole quotients.

Power

An expression formed by raising a base to an exponent.

The power 13^3 describes three repeated factors of thirteen.

Representation

A mathematically equivalent way to write a quantity.

8^4 is a representation of 4,096 using base eight.

Lesson preview

Slide 1 · The base is part of the instructionOpening · 18 min

When a question specifies the base, factor the number using that base repeatedly. A correct answer must satisfy both requirements: the displayed base is prescribed and its exponent reproduces the given quantity exactly.

• Read the requested base before beginning any factoring.
• Divide repeatedly by that base while quotients remain whole.
• The number of divisions becomes the exponent.
• Multiplying the recorded factors verifies the final expression.

Worked example. Divide 4,096 by 8 repeatedly until reaching 1. Count four divisions. Write 4,096 = 8^4 and multiply the four factors to verify. Answer: 4,096 = 8^4.

Check question: What is $1,331$ expressed in index form with base $11$?

1. ${11}^3$
2. ${11}^2$
3. ${11}^4$
4. $3^{11}$

Show answer

${11}^3$. To express $1,331$ with base $11$, repeatedly divide $1,331$ by $11$. $1,331÷11=121$ $121÷11=11$ $11÷11=1$ There are $3$ divisions, so there are three factors of $11$. Therefore, $1,331={11}^3$.

Slide 2 · Index notation in local planningPractice · 22 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• Both slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 12 lessons in this sub-strand

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All 13 lessons in Indices and Logarithms

1. 1Reading Repeated Factors as IndicesExpress numbers in index form
2. 2Index Forms with a Specified BaseExpress numbers in index form
3. 3Discovering Product and Quotient LawsDerive the laws of indices using factors
4. 4Power of a Power and Identity CasesDerive the laws of indices using factors
5. 5Indices in Measurement and DesignApply the laws of indices in different situations
6. 6From Powers of Ten to Common LogsRelate index notation to logarithm notation to base 10
7. 7Interpreting Base-Ten LogarithmsRelate index notation to logarithm notation to base 10
8. 8Reading Common Logarithms in TablesDetermine common logarithms of numbers from mathematical tables and calculators
9. 9Comparing Table and Calculator LogarithmsDetermine common logarithms of numbers from mathematical tables and calculators
10. 10Logarithms for Multiplication and DivisionApply common logarithms in multiplication, division, powers and roots of numbers
11. 11Logarithms for Powers and RootsApply common logarithms in multiplication, division, powers and roots of numbers
12. 12Integrated Logarithmic ComputationApply common logarithms in multiplication, division, powers and roots of numbers
13. 13Why Indices and Logarithms MatterAppreciate the use of indices and common logarithms in mathematical computations