Grade 10 · Core Mathematics · Lesson 4 of 13

Power of a Power and Identity Cases

A Grade 10 Core Mathematics CBE lesson from the sub-strand Indices and Logarithms. In it, learners derive the power-of-a-power rule from repeated factors.

Strand Numbers and AlgebraSub-strand Indices and LogarithmsLesson code 10CM-12-L440 minutes · 3 slides

KICD learning outcome (b)

Derive the laws of indices using factors.

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

Derive the power-of-a-power rule from repeated factors.
Recognise zero and negative exponents through division reasoning.
Explain each result using an equivalent factor argument.

Key vocabulary

Power of a power: An expression where one power is raised to another.; (3²)⁴ has four repeated blocks of two threes.
Outer exponent: The exponent applied to an entire parenthesised power.; In (5³)², two is the outer exponent.
Zero exponent: The exponent that gives one for every nonzero base.; 10⁰ equals one because no factors remain after cancellation.
Negative exponent: An exponent indicating the reciprocal of a positive power.; 2⁻³ equals 1 divided by 2³.
Reciprocal: A number formed by interchanging numerator and denominator.; The reciprocal of 8 is 1/8.
Identity: A value that leaves another value unchanged under an operation.; One is the multiplicative identity for nonzero numbers.

Lesson preview

Slide 1 · A power raised to another powerOpening · 12 min

Figure 1: A grid showing three groups of two fours, demonstrating that raising four squared to the power of three equals four to the power of six.

A power of a power repeats an entire factor string. If a base already has one exponent and that expression is repeated, the total number of base factors is found by multiplying the two counts. Study Figure 1 to see how multiplying the exponents gives the total number of base factors.

The inner exponent counts factors in one block.
The outer exponent counts repeated blocks of the inner factor string.
Multiplying exponent counts gives the total factors.
Parentheses show which power receives the outer exponent.

Worked example. Identify inner exponent 2 and outer exponent 3. Multiply exponents: 2 × 3 = 6. Write (4²)³ = 4⁶. Answer: 4⁶.

Check question: What is $(2^{3})^{2}$ expressed as a single power?

$2^{5}$
$2^{6}$
$2^{9}$
$4^{6}$

Show answer

$2^{6}$ . A power of a power repeats an entire factor string. The inner exponent ( $3$ ) counts factors in one block, and the outer exponent ( $2$ ) counts the repeated blocks. Multiplying the exponent counts ( $3 \times 2$ ) gives $6$ total factors of $2$ . Therefore, $(2^{3})^{2} = 2^{6}$ .

Slide 2 · Zero and negative exponents from quotientsModel · 14 min

Figure 2: A sequence diagram illustrating how repeatedly dividing by eleven decreases the exponent by one, leading to zero and negative powers.

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

All 3 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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Power of a Power and Identity Cases

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 13 lessons in Indices and Logarithms