Step-by-Step Examples of the Power of a Power Rule for Students
The concept of Power of a Power Rule is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding Power of a Power Rule
A Power of a Power Rule refers to an exponent law used when a number (base) is raised to a power, and that entire result is again raised to another power. This concept is widely used in algebraic expressions, simplifying exponents, and solving exponential equations. The rule is a key part of exponent laws taught in school and board-level exams.
Formula Used in Power of a Power Rule
The standard formula is: \( (a^m)^n = a^{m \times n} \)
Here’s a helpful table to understand Power of a Power Rule more clearly:
Power of a Power Rule Table
| Expression | Expanded | Simplified |
|---|---|---|
| (x2)3 | x2×3 | x6 |
| (34)2 | 34×2 | 38 |
| (a-3)2 | a-3×2 | a-6 |
This table shows how the pattern of combining exponents applies regularly in real cases with the Power of a Power Rule.
Step-by-Step Explanation of Power of a Power Rule
Let’s see how to use the Power of a Power Rule with actual steps:
1. Identify the base (a) and the two exponents (m and n) in the form (am)n.
2. Multiply the exponents: Calculate m × n.
3. Write the result as a single exponent with the base unchanged: \( a^{m \times n} \).
4. If possible, expand or further simplify the result.
Worked Examples – Power of a Power Rule
Let’s work through a few examples, including negative exponents and rational exponents:
Example 1: Simplify (x4)3.
1. Identify the exponents: m = 4, n = 3.
2. Multiply: 4 × 3 = 12.
3. Write the answer: x12.
Example 2: Simplify (2-2)5.
1. Exponents: m = -2, n = 5.
2. Multiply: -2 × 5 = -10.
3. Write the answer: 2-10 = 1 / 210.
Example 3: Simplify [(x + y)1/2]4.
1. Exponents: m = 1/2, n = 4.
2. Multiply: (1/2) × 4 = 2.
3. Answer: (x + y)2 = x2 + 2xy + y2.
Power of a Power Rule with Negative Exponents
You can apply the rule even when the exponents are negative. For example:
Example: Simplify (a-3)-2.
1. Exponents: m = -3, n = -2.
2. Multiply: -3 × -2 = 6.
3. Write the answer: a6.
Power of a Power Rule with Fractional Exponents
Example: Simplify (42/3)3/2.
1. Exponents: m = 2/3, n = 3/2.
2. Multiply: (2/3) × (3/2) = 1.
3. Answer: 41 = 4.
Proof and Explanation
The proof of the Power of a Power Rule comes from repeated multiplication:
For (am)n, you multiply am by itself n times.
That is, (am)n = am × am × ... × am (n times).
By the product of powers rule, this gives am + m + ... + m (n times) = am×n.
Practice Problems
- Simplify (y5)4
- Evaluate (5-2)3
- Find the value of (x3/4)8
- Simplify (2-1)-2
Common Mistakes to Avoid
- Confusing Power of a Power Rule with the product of powers law.
- Forgetting to multiply, instead of adding, the exponents.
- Ignoring negative exponents or mishandling the signs.
- Applying the rule when the bases are not the same.
Related Exponent Rules
| Rule | Formula | When to Use |
|---|---|---|
| Product of Powers | am × an = am+n | When bases are same and multiplied |
| Quotient of Powers | am ÷ an = am-n | When dividing same bases |
| Power of a Power | (am)n = am×n | Base raised to a power, then to another power |
| Negative Exponent | a-n = 1/an | When exponent is negative |
For further details, review the Laws of Exponents and Exponents pages for more examples and explanations.
Real-World Applications
The Power of a Power Rule appears in science (energy calculations), finance (compound interest), and computer science (data storage). Vedantu helps students see how these principles are applied beyond the classroom for real-life success.
We explored the idea of Power of a Power Rule, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Suggested In-Depth Resources
- Laws of Exponents
- Exponents
- Negative Exponents
- Product of Powers Law
- Quotient of Powers Law
- Scientific Notation
- Powers and Exponents Worksheet
- Powers
FAQs on Understanding the Power of a Power Rule in Math
1. What is the power of a power rule in exponents?
The power of a power rule in exponents states that when you raise a power to another power, you multiply the exponents. In mathematical terms, (am)n = am×n. This rule is commonly used to simplify expressions with exponents.
2. How do you calculate a power to a power?
To calculate a power to a power, use the power of a power rule: (am)n = am×n.
For example, (23)4 = 212, since 3 × 4 = 12.
3. What are some examples of the power of a power rule?
Examples using the power of a power rule:
• (32)3 = 36 (because 2 × 3 = 6)
• (x5)2 = x10 (because 5 × 2 = 10)
• (a-4)3 = a-12 (because -4 × 3 = -12)
4. What are the 7 rules of exponents?
The seven rules of exponents are:
1. Product Rule: am × an = am+n
2. Quotient Rule: am ÷ an = am-n
3. Power of a Power Rule: (am)n = am×n
4. Power of a Product Rule: (ab)n = anbn
5. Power of a Quotient Rule: (a/b)n = an/bn
6. Zero Exponent Rule: a0 = 1 (for a ≠ 0)
7. Negative Exponent Rule: a-n = 1/an
5. How to solve power rule step by step?
To solve using the power of a power rule:
1. Identify the base and the exponents.
2. Multiply the two exponents.
3. Rewrite the expression with the new exponent.
Example: (x4)3 → x4×3 → x12.
6. Can you explain the power of a power rule with negative exponents?
Power of a power rule works the same with negative exponents:
(am)n = am×n, even if m or n are negative.
Example: (5-2)3 = 5-6 (as -2 × 3 = -6).
7. What is the difference between power of a product and power of a power rule?
Power of a product rule states that (ab)n = anbn, while power of a power rule deals with exponents on exponents: (am)n = am×n. Both are used to simplify powers but apply to different forms.
8. What is the power of a quotient rule with examples?
Power of a quotient rule states (a/b)n = an/bn.
Example: (2/3)4 = 24/34 = 16/81.
9. How do you use a power of a power rule calculator?
To use a power of a power rule calculator, input the base number and both exponents. The tool multiplies the exponents and shows the result as a single exponent: (am)n = am×n. Many educational websites have these calculators for practice and verification.
10. How can the power of a power rule be proved?
The power of a power rule can be proved by expanding exponents:
(am)n means multiplying am by itself n times.
So, (am)n = am × am × ... (n times) = am×n (using the product rule of exponents).
11. Where is the power of a power rule used in real-life problems?
The power of a power rule is used in real-life calculations such as scientific notation, compound interest, population growth, and whenever repeated multiplication of factors occurs, especially in physics, chemistry, and financial mathematics.
12. Can you solve example problems using the power of a power rule?
Sure! Here are some examples:
• (x3)4 = x12
• (2-2)5 = 2-10
• (a1)0 = a0 = 1
These use the rule (am)n = am×n to simplify exponents.
