Table Of Contents

What is the Formula for Power of a Power Rule?

What is the Power of a Power Rule With Negative Exponents?

What is the Fraction Power to Power Rule?

How to Simplify Expressions in the Power of a Power Rule?

Real-life Applications of Power of a Power Rule

Common Mistakes and How to Avoid Them in the Power of a Power Rule

Solved Examples of Power of a Power Rule

FAQs on Power of a Power Rule

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Last updated on July 17th, 2025

Power of a Power Rule

Q: What is the power of a power rule?

Expressions of the form ((xa)b) can be simplified with the help of the power of a power rule.

Q: What is the formula of the power of a power rule?

The formula of the power of a power rule is: (ax)y = axy.

Q: What are the laws of exponents?

The laws of exponents are: am × an = am + n am/an = am - n (am)n = amn a0 = 1 a-m = 1/am

Q: Find the value of (5^2)^5.

The value of (52)5 is 52 × 5 = 510 = 9765625

Q: What is the formula of a power of a power rule for negative exponents?

The formulas, when negative exponents are used, are given below: (a-m)-n = amn (a-m)n = a-mn (am)-n = a-mn

Q: How does learning Algebra help students in Canada make better decisions in daily life?

Algebra teaches kids in Canada to analyze information and predict outcomes, helping them in decisions like saving money, planning schedules, or solving problems.

Q: How can cultural or local activities in Canada support learning Algebra topics such as Power of a Power Rule ?

Traditional games, sports, or market activities popular in Canada can be used to demonstrate Algebra concepts like Power of a Power Rule , linking learning with familiar experiences.

Q: How do technology and digital tools in Canada support learning Algebra and Power of a Power Rule ?

At BrightChamps in Canada, we encourage students to use apps and interactive software to demonstrate Algebra’s Power of a Power Rule , allowing students to experiment with problems and see instant feedback for better understanding.

Q: Does learning Algebra support future career opportunities for students in Canada?

Yes, understanding Algebra helps students in Canada develop critical thinking and problem-solving skills, which are essential in careers like engineering, finance, data science, and more.

Professor Greenline Explaining Math Concepts

In algebra, there are laws to simplify expressions. Power of a power rule is used to work with bases where one exponent is raised to another, like ((x^a)^b). In this article, we will discuss the power of the power rule in detail.

Power of a Power Rule for Canadian Students

What is the Power of a Power Rule?

The power of a power rule is among the most important exponent laws. It is mainly applied to simplify expressions in the form (x^a)^b. Mathematically, it can be represented as (x^a)^b = x^{a × b} = x^ab where the exponents are multiplied together.

What is the Formula for Power of a Power Rule?

The formula for the power of a power rule is (x^a)^b = x^ab where x is the base, and a and b are exponents. This formula is used to solve expressions like:

(x³)² = x^{3 × 2} = x⁶
(5⁵)³ = 5^{5 × 3} = 5¹⁵
(x⁴)³ = x^{4 × 3} = x¹²

What is the Power of a Power Rule With Negative Exponents?

The same rule is applied even for expressions with negative exponents. In (x^a)^b, if a and b are less than 0, then both the exponents are negative. Therefore, the formulas will change accordingly:

(a^-m)^-n = a^{(-m) × (-n)} = a^mn
(a^-m)ⁿ = a^{(-m) × (n)} = a^-mn
(a^m)^-n= a^{(m) × (-n)} = a^-mn

What is the Fraction Power to Power Rule?

If the exponents are in the fractional form of p/q, where p and q are integers, then we can use the formula ((a^p/q)^m/n) to solve such expressions. Let us take a look at the formulas when the exponents are fractions:

(x^m/n)^p/q = x^mp/nq
(x^m)^p/q = x^pm/q
(x^m/n)^p = x^pm/n

How to Simplify Expressions in the Power of a Power Rule?

So far, we’ve learned about the power of a power rule. In this section, we will see how to simplify expressions using this rule.

For example, simplify (5²)³.

The formula of the power of a power rule is: (x^a)^b = x^ab

Here, x = 5, a = 2, and b = 3

Substituting the values we get, (5²)³ = 5^{(2 × 3)}

= 5⁶

= 5 × 5 × 5 × 5 × 5 × 5

= 15625

Real-life Applications of Power of a Power Rule

The objective of the power of a power rule is to simplify expressions with an exponent raised to another exponent. Here are some real-life applications:

Finance: As the formula for calculating compound interest is A = P(1 + (r/n))^nt, we use the power of the power rule in banks and financial institutions.

Physics: It is used to represent very large or very small numbers in astronomy, physics, or nanotechnology. For example, (5 × 10¹⁸)² can be written as 5 × 10³⁶. Scientific notations like 5 × 1036 are used in various scientific fields.

Computing Power: In computer science, the rule is used to calculate nested exponential growth in computing.

Common Mistakes and How to Avoid Them in the Power of a Power Rule

When using the power of a power rule, students make errors by either confusing it with other mathematical rules or misapplying it. This section talks about some of the mistakes that can be avoided.

Mistake 1

Adding the Exponents

While adding the exponents of the number, the students often make mistakes in adding the exponents instead of multiplying the power.

For example, (5²)³ = 5^{2 + 3} = 5⁵ instead of (5²)³ = 5^{2 × 3} = 5⁶. So, always remember that the power of a power rule states that (a^x)^y = a^xy.

Mistake 2

Confusing Exponents With the Base

Many students get confused between exponents and base.

For example, expression like (x²)³, some may mistakenly assume 2 and 3 to be the base. However, in the given expression, x is the base and 2 and 3 are exponents.

Mistake 3

Making Mistakes While Dealing With Negative Exponents

When applying the power of a power rule with negative exponents, students can make errors as they may get confused with the sign.

For example, while computing (2^-2)^-3, some students may mistakenly write the answer as 2^-6, whereas the correct answer is 2⁶. Memorizing the formulas and the sign rules correctly will alleviate confusion.

Mistake 4

Not Simplifying the Expression

Students apply the power of a power rule, but they do not simplify it, for example, (3x²)⁴ = (3⁴)(x²)⁴ = 3⁴x⁸ instead of 81x⁸. So, always simplify the exponent multiplication.

Mistake 5

Ignoring Zero Exponents

When using the power of power rule for zero exponents, students think that a zero exponent means zero.

For example, (x⁰)⁴ = 0⁴, which is wrong as (x⁰)⁴ = 1⁴ = 1. Always remember that a number raised to 0 is 1, x⁰ = 1.

Solved Examples of Power of a Power Rule

Ray, the Character from BrightChamps Explaining Math Concepts

Problem 1

Find the value of (5^3)^4?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The value of (5³)⁴ is 244140625

Explanation

We find the value of (5³)⁴ using the formula:

(x^a)^b = x^ab

So, (5³)⁴ = 5^{3 × 4}

= 5¹²

= 244140625

Problem 2

Find the value of ((-2 + 3)^2)^5?

The value of ((-2 + 3)2)5 is 1

Explanation

The first step is to solve the inner parentheses.

-2 + 3 = 1

Now, ((-2 + 3)²)⁵ = (1²)⁵

(1²)⁵ is of the form (x^a)^b which can be written as x^ab

(1²)⁵ = 1^{2 × 5}

= 1¹⁰

= 1

Problem 3

Find the value of (5^-2)^-3?

The value of (5^-2)^-3 is 15625

Explanation

The value of (5^-2)^-3 can be found using the power of a power rule.

That is (a^-m)^-n = a^mn

(5^-2)^-3 = 5^{-2 × -3}

= 5⁶

= 15625

Problem 4

Simplify: (x^2)^6?

x¹²

Explanation

(x²)⁶can be simplified by keeping the base and multiplying only the exponents.

(x²)⁶ = x¹²

Problem 5

Find the value of ((-5)^-2)^-3?

The value of ((-5)^-2)^-3 is 15625

Explanation

Multiplying the exponents: -2 × -3 = 6

So, ((-5)^-2)^-3 = (-5)⁶

= -15625

FAQs on Power of a Power Rule

1.What is the power of a power rule?

2.What is the formula of the power of a power rule?

3.What are the laws of exponents?

4.Find the value of (5^2)^5.

5.What is the formula of a power of a power rule for negative exponents?

6.How does learning Algebra help students in Canada make better decisions in daily life?

7.How can cultural or local activities in Canada support learning Algebra topics such as Power of a Power Rule ?

8.How do technology and digital tools in Canada support learning Algebra and Power of a Power Rule ?

9.Does learning Algebra support future career opportunities for students in Canada?

Explore More algebra

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.