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Power Of a Power Rule
The power of a power rule in exponents is a rule that is applied to simplify an algebraic expression when a base is raised to a power, and then the whole expression is raised to another power. Before we get into the detail of the concept, let us recall the meaning of power and base. For the expression b^{x}, b is the base and x is the power (also called the exponent) which implies b is multiplied by itself x times. Now, the power of a power rule is used to simplify expressions of the form (b^{x})^{y} which on simplification is written as b^{xy}. To apply power to the power rule, we multiply the two powers keeping the base the same.
Further, in this article, we will explore the power to the power rule in detail and its formula. We will understand the application of the power of a power rule in the simplification of algebraic expressions with negative and rational exponents. We will solve a few examples based on the concept for a better understanding.
What is Power of a Power Rule?
The power of a power rule in exponents when a base is raised to a power and the whole expression is again raised to another power, that is, when we have an expression of the form (a^{m})^{n} as here 'a' is the base which is raised to the power 'm' and then the whole expression a^{m} is raised to another power 'n'. To simplify this, we use the power to the power rule given by, (a^{m})^{n} = a^{m}^{n}, where we multiply the two powers 'm' and 'n' keeping the base the same as 'a'. We can state the power to the power rule as 'If the base raised to a power is being raised to another power, then the two powers are multiplied and the base remains the same.'
Power To the Power Rule Formula
The formula for the power to the power rule is given by, (a^{m})^{n} = a^{m}^{n}, where a is the base and m, n are the powers, is given by, (a^{m})^{n} = a^{m}^{n}. We apply this formula when an exponent is given in the form (a^{m})^{n}. We can simply multiply the powers and keep the base the same. Some of the examples of the rule are:
 (x^{2})^{3} = x^{2×3} = x^{6}
 (3^{4})^{2} = 3^{4×2} = 3^{8}
 [(x + y)^{5}]^{7} = (x + y)^{5×7} = (x + y)^{35}
Power Of a Power Rule With Negative Exponents
Now, we know the formula for the power to the power rule. When the power of the base is negative, we can apply the same formula by multiplying the exponents. So, if m > 0 and n > 0 and we have negative exponents, then using the same formula as given above, we have
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
Using the abovegiven formulas, we can apply the power of a power rule and simplify expressions with negative exponents.
Fraction Power To Power Rule
Fraction powers are those when the exponents of a base are of the form p/q, where p and q are integers. So, we apply the same formula of the power to the power rule to simplify the expression. So, the formula for the rational power of a power rule is given by, (a^{p/q})^{m/n} = a^{pq/mn}. Here, we multiply the two numerators and the two denominators separately. Some of the examples of rational power of a power rule are:
 (x^{1/3})^{2} = x^{2/3}
 (4^{3}^{/2})^{2/3} = 4^{3×2/2×3} = 4^{1} = 4
 (2^{2})^{3/2} = 2^{2 × 3/2} = 2^{3} = 1/2^{3}
Simplifying Power Of a Power Rule
Now that we know the formula for the power to the power rule with positive exponents, negative exponents, and rational exponents. Let us solve a few examples and apply the formula to understand its application.
Example 1: Find the value of (2^{2})^{5}.
Solution: To simplify the expression (2^{2})^{5}, we apply the power to the power rule and multiply the powers 2 and 5.
(2^{2})^{5} = (2)^{2×5}
= (2)^{10}
= 2^{10}  [Because the power 10 is even]
= 1024
Example 2: Simplify the expression (x^{5})^{9}
Solution: We can observe that the expression (x^{5})^{9} has a negative power. So, we multiply the two powers 5 and 9 to obtain the result and keep the base x the same.
(x^{5})^{9} = x^{5×9}
= x^{45}
Example 3: Evaluate the value of (3^{2/3})^{3/4}.
Solution: To find the value of (3^{2/3})^{3/4}, we will use the power of a power rule for rational exponents. We will simply multiply the powers 2/3 and 3/4 keeping the base the same as 3. So, we have
(3^{2/3})^{3/4} = 3^{2/3×3/4}
= 3^{2/4}
= 3^{1/2}
= 1/√3  [Using exponent rule a^{m} = 1/a^{m}]
Important Notes on Power of a Power Rule
 The power to the power rule states that 'If the base raised to a power is being raised to another power, then the two powers are multiplied and the base remains the same.'
 The formula for the power of a power rule is (a^{m})^{n} = a^{m}^{n}.
 Power of a power rule for negative exponents:
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
 Rational power to the power rule: (a^{p/q})^{m/n} = a^{pq/mn}
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Power Of a Power Rule Examples

Example 1: Simplify the expression [(x + y)^{1/2}]^{4} using the power of a power rule.
Solution: To simplify the algebraic expression [(x + y)^{1/2}]^{4}, we will multiply the power 1/2 and 4 and keep the base the same as x + y. So, we have
[(x + y)^{1/2}]^{4 }= (x + y)^{(1/2) × 4}
= (x + y)^{4/2}
= (x + y)^{2}
= x^{2} + 2xy + y^{2}  [Using algebraic identity (a + b)^{2} = a^{2} + 2ab + b^{2}]
Answer: Thus, [(x + y)^{1/2}]^{4} = x^{2} + 2xy + y^{2}

Example 2: Calculate the value of (10^{3})^{7}
Solution: To find the value of (10^{3})^{7}, we will use the power to the power rule. We will simply multiply the powers 3 and 7. So, we have
(10^{3})^{7} = 10^{3×}^{7}
= 10^{21}
Answer: Thus the value of (10^{3})^{7} is equal to 10^{21}.

Example 3: What is the value [(1 + 2)^{2}]^{7/8}?
Solution: We will simplify the expression [(1 + 2)^{2}]^{7/8} using the power of a power rule. We have
[(1 + 2)^{2}]^{7/8} = (1)^{2×7/8}
= (1)^{14/8}
= (1)^{7/4}
= 1
Answer: So, [(1 + 2)^{2}]^{7/8} is equal to 1.
FAQs on Power Of a Power Rule
What is Power of a Power Rule in Math?
The power of a power rule in exponents is a rule that is applied to simplify an algebraic expression when a base is raised to a power, and then the whole expression is raised to another power. The rule states that 'If the base raised to a power is being raised to another power, then the two powers are multiplied and the base remains the same.'
What is the Formula of Power to the Power Rule?
The formula for the power to the power rule is given by, (a^{m})^{n} = a^{m}^{n}, where a is the base and m, n are the powers, is given by, (a^{m})^{n} = a^{m}^{n}. We can simply multiply the powers and keep the base the same.
What is Power of a Power Rule for Negative Exponents?
When the power of the base is negative, we can apply the same formula (a^{m})^{n} = a^{m}^{n} by multiplying the exponents. If m > 0 and n > 0, then we have
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
 (a^{m})^{n} = a^{m×n} = a^{mn}
How to Simplify Algebraic Expressions With Rational Exponents Using Power of a Power Rule?
The formula for the rational power of a power rule is given by, (a^{p/q})^{m/n} = a^{pq/mn}. We simply multiply the rational exponents to apply the rational power of a power rule.
How Do You Apply the Power To the Power Rule?
To apply the power to the power rule, we simply multiply the powers keeping the base the same, and obtain the result. If we have (a^{m})^{n}, then we have two powers m and n. Here, we will just multiply the powers m and n and keep the base the same. So, we have (a^{m})^{n} = a^{m}^{n}.
What is Fraction Power To Power Rule?
The fraction power to power rule is applied when the powers are expressed as fractions. Its formula is given by, (a^{p/q})^{m/n} = a^{pq/mn}
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