Apply the Power of a Power Property

Oct 13, 2025 by magicalceylontours.com 54 views

Introduction

Exponential expressions are a fundamental concept in mathematics, and solving them requires a deep understanding of various mathematical identities. In this article, we will explore two exponential expressions and use identities to simplify them. We will start by understanding the properties of exponents and then apply these properties to solve the given expressions.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be written as $2 \times 2 \times 2$ . Exponents can be added, subtracted, multiplied, and divided using the following properties:

Product of Powers: $a^m \times a^n = a^{m+n}$
Power of a Power: $(a^m)^n = a^{m \times n}$
Power of a Product: $(ab)^m = a^m \times b^m$
Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

Solving Expression (a)

The first expression we need to solve is $2^{15 / 5 \cdot 2^{21 / 5}}$ . To simplify this expression, we can use the property of exponents that states $(a^m)^n = a^{m \times n}$ .

Step 1: Simplify the Exponent

We can start by simplifying the exponent $15 / 5 \cdot 2^{21 / 5}$ . Using the property of exponents, we can rewrite this as $3 \cdot 2^{21 / 5}$ .

import math

# Define the variables
a = 2
m = 15 / 5
n = 21 / 5

# Simplify the exponent
exponent = m * (a ** n)

Step 2: Apply the Power of a Power Property

Now that we have simplified the exponent, we can apply the power of a power property to rewrite the expression as $2^{3 \cdot 2^{21 / 5}}$ .

# Apply the power of a power property
expression = a ** (3 * (a ** n))

Step 3: Simplify the Expression

Finally, we can simplify the expression by evaluating the exponent. Using the property of exponents, we can rewrite this as $2^{3 \cdot 2^{21 / 5}} = 2^{3 \cdot 2^{2.2}}$ .

# Simplify the expression
result = 2 ** (3 * (2 ** 2.2))

Solving Expression (b)

The second expression we need to solve is $25^{25 / 50}$ . To simplify this expression, we can use the property of exponents that states $(a^m)^n = a^{m \times n}$ .

Step 1: Simplify the Exponent

We can start by simplifying the exponent $25 / 50$ . Using the property of exponents, we can rewrite this as $5 / 10$ .

# Define the variables
a = 25
m = 25 / 50

# Simplify the exponent
exponent = m

Step 2: Apply the Power of a Power Property

Now that we have simplified the exponent, we can apply the power of a power property to rewrite the expression as $5^{5 / 10}$ .

# Apply the power of a power property
expression = 5 ** exponent

Step 3: Simplify the Expression

Finally, we can simplify the expression by evaluating the exponent. Using the property of exponents, we can rewrite this as $5^{5 / 10} = 5^{0.5}$ .

# Simplify the expression
result = 5 ** 0.5

Conclusion

In this article, we have used identities to simplify two exponential expressions. We have applied the properties of exponents, including the product of powers, power of a power, power of a product, and quotient of powers. By simplifying the exponents and applying the power of a power property, we have been able to rewrite the expressions in a simpler form. This demonstrates the importance of understanding the properties of exponents in solving exponential expressions.

References

Introduction

In our previous article, we explored two exponential expressions and used identities to simplify them. We applied the properties of exponents, including the product of powers, power of a power, power of a product, and quotient of powers. In this article, we will answer some frequently asked questions about solving exponential expressions using identities.

Q: What are the properties of exponents?

A: The properties of exponents are:

Product of Powers: $a^m \times a^n = a^{m+n}$
Power of a Power: $(a^m)^n = a^{m \times n}$
Power of a Product: $(ab)^m = a^m \times b^m$
Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

Q: How do I simplify an exponential expression using identities?

A: To simplify an exponential expression using identities, you can follow these steps:

Simplify the Exponent: Simplify the exponent by combining like terms and applying the properties of exponents.
Apply the Power of a Power Property: Apply the power of a power property to rewrite the expression in a simpler form.
Simplify the Expression: Simplify the expression by evaluating the exponent.

Q: What is the difference between $2^{15 / 5 \cdot 2^{21 / 5}}$ and $25^{25 / 50}$ ?

A: The two expressions are different because they have different bases and exponents. The first expression has a base of 2 and an exponent of $15 / 5 \cdot 2^{21 / 5}$ , while the second expression has a base of 25 and an exponent of $25 / 50$ .

Q: How do I evaluate the exponent in an exponential expression?

A: To evaluate the exponent in an exponential expression, you can use the properties of exponents. For example, if you have an expression like $2^{3 \cdot 2^{21 / 5}}$ , you can evaluate the exponent by applying the power of a power property and simplifying the expression.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

Not simplifying the exponent: Failing to simplify the exponent can lead to incorrect results.
Not applying the power of a power property: Failing to apply the power of a power property can lead to incorrect results.
Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.

Q: How do I check my work when simplifying exponential expressions?

A: To check your work when simplifying exponential expressions, you can:

Plug in values: Plug in values for the variables in the expression to see if the result is correct.
Use a calculator: Use a calculator to evaluate the expression and see if the result is correct.
Check the properties of exponents: Check the properties of exponents to make sure you are applying them correctly.

Conclusion

Solving exponential expressions using identities requires a deep understanding of the properties of exponents. By simplifying the exponent, applying the power of a power property, and simplifying the expression, you can rewrite exponential expressions in a simpler form. Remember to check your work by plugging in values, using a calculator, and checking the properties of exponents. With practice and patience, you can become proficient in solving exponential expressions using identities.