Solved Exercises on the Power of a Power Rule in Grade 7

Master the power of a power rule: base, exponent, nested powers, through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Power of Power
Exercise 1
Calculate: \( (2^3)^2 \)
Definition:

Power of a power: An expression of the form \((a^m)^n\) where you raise a power to another exponent

Power of power method:
  1. Multiply the exponents together
  2. Keep the same base
  3. Apply the rule: \((a^m)^n = a^{mn}\)
Original Expression
\((2^3)^2\)
Identify Components
Base = 2, Inner exp = 3, Outer exp = 2
Apply Rule
\((a^m)^n = a^{mn}\)
Multiply Exponents
\(3 \times 2 = 6\)
Result
\(2^6\)
Step 1: Identify the structure

We have \((2^3)^2\) where base is 2, inner exponent is 3, outer exponent is 2

Step 2: Apply the power of a power rule

\((a^m)^n = a^{m \times n}\)

\((2^3)^2 = 2^{3 \times 2}\)

Step 3: Multiply the exponents

\(3 \times 2 = 6\)

Step 4: Write the result

\(2^6\)

Step 5: Calculate if needed

\(2^6 = 64\)

\( (2^3)^2 = 2^6 = 64 \)
Final answer:

\( (2^3)^2 = 2^6 \) or \(64\)

Applied rules:

• Power of a power: \((a^m)^n = a^{mn}\)

• Base remains: Keep the original base unchanged

• Exponent multiplication: Multiply the exponents together

2 Negative Base
Exercise 2
Calculate: \( (-3^2)^3 \)
Definition:

Order of operations: Handle exponents before the negative sign unless parentheses indicate otherwise

Original Expression
\((-3^2)^3\)
Evaluate Inside First
\(3^2 = 9\), so \(-9\)
Apply Outer Power
\((-9)^3\)
Calculate
\(-729\)
Step 1: Identify the order of operations

Inside parentheses: \(-3^2\). Exponent applies to 3 first, then apply negative

Step 2: Evaluate the inner expression

\(3^2 = 9\), so \(-3^2 = -9\)

Step 3: Apply the outer power

\((-9)^3 = (-9) \times (-9) \times (-9)\)

Step 4: Calculate step by step

\((-9) \times (-9) = 81\)

\(81 \times (-9) = -729\)

Step 5: Write the final answer

\(-729\)

\( (-3^2)^3 = -729 \)
Final answer:

\( (-3^2)^3 = -729 \)

Applied rules:

• Order of operations: Exponents before negation (PEMDAS)

• Odd exponent: Negative base to odd power = negative result

• Sign multiplication: Negative × Negative = Positive, Positive × Negative = Negative

3 Fractional Base
Exercise 3
Calculate: \( \left(\frac{2}{3}\right)^2 \)
Definition:

Power of a fraction: Raise both numerator and denominator to the power: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Original Expression
\(\left(\frac{2}{3}\right)^2\)
Apply Fraction Rule
\(\frac{2^2}{3^2}\)
Calculate Numerator
\(2^2 = 4\)
Calculate Denominator
\(3^2 = 9\)
Result
\(\frac{4}{9}\)
Step 1: Identify the fraction and exponent

Base is \(\frac{2}{3}\) and exponent is 2

Step 2: Apply the fraction power rule

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

\(\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2}\)

Step 3: Calculate numerator

\(2^2 = 2 \times 2 = 4\)

Step 4: Calculate denominator

\(3^2 = 3 \times 3 = 9\)

Step 5: Write the final fraction

\(\frac{4}{9}\)

\( \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)
Final answer:

\( \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)

Applied rules:

• Fraction power rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

• Separate powers: Apply exponent to numerator and denominator independently

• Positive result: Positive fraction to any power remains positive

Rules and methods, laws,...
\( (a^m)^n = a^{mn} \)
Power of a Power Rule
Basic Rule
\( (a^m)^n = a^{mn} \)
Multiply exponents, keep base
With Variables
\( (x^a)^b = x^{ab} \)
Works with variable bases
Fractional Base
\( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)
Apply exponent to both parts
Key definitions:

Power: An expression of the form \(a^n\) where \(a\) is the base and \(n\) is the exponent

Base: The number that is being multiplied repeatedly

Exponent: The number of times the base is multiplied by itself

Power of a power: An expression where a power is raised to another exponent

Nested exponents: Exponents within exponents

Order of operations: Rules for the sequence of calculations (PEMDAS/BODMAS)

Power of power methods:
  1. Identify structure: Recognize \((a^m)^n\) format
  2. Multiply exponents: Multiply the inner and outer exponents
  3. Keep base: The base remains unchanged
  4. Simplify: Calculate if needed
Tip 1: Always multiply the exponents, don't add them
Tip 2: Pay attention to parentheses - they determine what's in the base
Tip 3: The rule works with positive, negative, and fractional bases
Tip 4: Verify by expanding to check your answer
Common errors: Adding exponents instead of multiplying, misidentifying the base, not following order of operations.
Exam preparation: Practice with various base types, memorize the rule, watch for sign patterns.
Formulas to know by heart:

• Basic rule: \((a^m)^n = a^{mn}\)

• Fraction rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

• Zero exponent: \((a^0)^n = 1\) (where \(a \neq 0\))

• Identity: \((a^1)^n = a^n\)

Solution: Exercises 4 to 5
4 Variable Base
Exercise 4
Calculate: \( (x^4)^3 \) where \(x \neq 0\)
Definition:

Variable base: The power rule works with variables: \((x^a)^b = x^{ab}\)

Original Expression
\((x^4)^3\)
Identify Components
Base = x, Inner exp = 4, Outer exp = 3
Apply Rule
\((x^a)^b = x^{ab}\)
Multiply Exponents
\(4 \times 3 = 12\)
Result
\(x^{12}\)
Step 1: Identify the variable base and exponents

Base is \(x\), inner exponent is 4, outer exponent is 3

Step 2: Apply the power of a power rule

\((x^m)^n = x^{mn}\)

\((x^4)^3 = x^{4 \times 3}\)

Step 3: Multiply the exponents

\(4 \times 3 = 12\)

Step 4: Write the final answer

\(x^{12}\)

Step 5: Note the condition

Condition: \(x \neq 0\) to avoid division by zero in related problems

\( (x^4)^3 = x^{12} \)
Final answer:

\( (x^4)^3 = x^{12} \)

Applied rules:

• Power of power: Works with variable bases

• Exponent multiplication: Multiply the exponents

• Variable preservation: Base variable remains unchanged

5 Multiple Variables
Exercise 5
Calculate: \( (2x^3y^2)^4 \)
Definition:

Power of a product: When raising a product to a power, raise each factor to that power: \((abc)^n = a^n b^n c^n\)

Original Expression
\((2x^3y^2)^4\)
Apply Power to Each Factor
\(2^4 \times (x^3)^4 \times (y^2)^4\)
Calculate Each Part
\(16 \times x^{12} \times y^8\)
Combine
\(16x^{12}y^8\)
Step 1: Identify all factors in the product

Factors are: 2, \(x^3\), and \(y^2\)

Step 2: Apply the power to each factor

\((2x^3y^2)^4 = 2^4 \times (x^3)^4 \times (y^2)^4\)

Step 3: Calculate each part separately

\(2^4 = 16\)

\((x^3)^4 = x^{3 \times 4} = x^{12}\)

\((y^2)^4 = y^{2 \times 4} = y^8\)

Step 4: Combine the results

\(16 \times x^{12} \times y^8 = 16x^{12}y^8\)

Step 5: Write the final answer

\(16x^{12}y^8\)

\( (2x^3y^2)^4 = 16x^{12}y^8 \)
Final answer:

\( (2x^3y^2)^4 = 16x^{12}y^8 \)

Applied rules:

• Power of a product: \((abc)^n = a^n b^n c^n\)

• Power of a power: Apply to each variable term

• Constant factor: Raise numerical coefficient to the power

Key Concepts: Laws, Methods, Rules, Definitions
\( (a^m)^n = a^{mn} \)
Fundamental Power of a Power Rule
Key definitions:

Power: An expression of the form \(a^n\) where \(a\) is the base and \(n\) is the exponent

Base: The number being multiplied repeatedly

Exponent: The number of times the base is multiplied by itself

Power of a power: An expression of the form \((a^m)^n\)

Exponent multiplication: The operation performed on nested exponents

Order of operations: Sequence of calculations (PEMDAS/BODMAS)

Complete power of power methodology:
  1. Identify structure: Recognize the \((a^m)^n\) pattern
  2. Extract components: Identify base, inner exponent, outer exponent
  3. Apply rule: Multiply the exponents
  4. Preserve base: Keep the original base unchanged
  5. Simplify: Calculate if needed
  6. Verify: Check with expansion if possible
Tip 1: Remember: "Multiply the exponents" - never add them
Tip 2: Pay attention to parentheses - they determine the base
Tip 3: Works for all number types: positive, negative, fractional
Tip 4: Always verify with expansion for small exponents
Common errors: Adding exponents instead of multiplying, misidentifying base due to missing parentheses, not following order of operations.
Exam preparation: Practice various base types, memorize the rule, watch for sign patterns with negative bases.
Formulas to know by heart:

• Basic rule: \((a^m)^n = a^{mn}\)

• Fraction rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

• Product rule: \((ab)^n = a^n b^n\)

• Zero exponent: \((a^0)^n = 1\) (where \(a \neq 0\))

• Identity: \((a^1)^n = a^n\)

Exercise with Visualization: Power of Power Patterns
Exercise 6: Nested Power Patterns
Observe these power of power patterns:
\( (2^2)^3 = 2^6 = 64 \)
\( (3^2)^2 = 3^4 = 81 \)
\( (5^1)^4 = 5^4 = 625 \)
\( (2^3)^2 = 2^6 = 64 \)

Analysis: The chart shows how nested exponents multiply.

  • \( (2^2)^3 = 2^{2 \times 3} = 2^6 = 64 \)
  • \( (3^2)^2 = 3^{2 \times 2} = 3^4 = 81 \)
  • \( (5^1)^4 = 5^{1 \times 4} = 5^4 = 625 \)
  • \( (2^3)^2 = 2^{3 \times 2} = 2^6 = 64 \)

Questions & Answers

Question: Why do we multiply the exponents when we have a power of a power? It seems like we should add them.

Answer: This is a great question! Let's look at why we multiply exponents in a power of a power:

Example: \((2^3)^2\)

  • First, \(2^3 = 2 \times 2 \times 2 = 8\)
  • Then, \((2^3)^2 = 8^2 = 8 \times 8 = 64\)
  • But we can also think of it as: \((2^3)^2 = (2 \times 2 \times 2)^2 = (2 \times 2 \times 2) \times (2 \times 2 \times 2)\)
  • This gives us 2 multiplied by itself \(3 \times 2 = 6\) times!

So \((2^3)^2 = 2^6\), which is the same as multiplying the exponents: \(3 \times 2 = 6\).

We're not adding exponents here because we're not combining like terms - we're nesting operations!

Question: What's the difference between \(-a^n\) and \((-a)^n\)? They look the same!

Answer: These expressions are completely different due to order of operations:

  • \(-a^n\): The exponent applies to \(a\) first, then the negative sign is applied
  • \((-a)^n\): The negative sign is part of the base, so everything is raised to the power

Examples:

  • \(-2^3 = -(2^3) = -8\) (exponent first, then negative)
  • \((-2)^3 = (-2) \times (-2) \times (-2) = -8\) (base includes negative)
  • \(-2^4 = -(2^4) = -16\) (exponent first, then negative)
  • \((-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16\) (even exponent makes positive)

The parentheses determine whether the negative is part of the base!

Question: How do I check if my power of a power calculation is correct?

Answer: Here are several ways to verify your power of a power calculations:

  1. Expand method: Write out the full multiplication to check
  2. Step verification: Ensure you multiplied exponents correctly
  3. Base check: Confirm the base remained unchanged
  4. Reasonableness: Does the answer seem appropriate for the numbers involved?

Example verification: For \((3^2)^3 = 3^6\)

  • Expand: \((3^2)^3 = 9^3 = 729\)
  • Rule: \(3^{2 \times 3} = 3^6 = 729\)
  • Check: Both methods give 729 ✓

Always double-check that you applied the rule to the correct base!