Use The Properties Of Exponents To Simplify The Expression:$\frac{\left(a^5\right)^3}{a^7}$

Introduction

Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for simplifying complex expressions. In this article, we will explore how to use the properties of exponents to simplify the expression $\frac{\left(a^5\right)^3}{a^7}$. We will delve into the rules of exponents, apply them to the given expression, and provide a step-by-step guide on how to simplify it.

The Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. The rules of exponents state that:

• Product of Powers: When multiplying two powers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers: When dividing two powers with the same base, subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's apply them to the given expression $\frac{\left(a^5\right)^3}{a^7}$. We will use the Power of a Power rule to simplify the numerator and the Quotient of Powers rule to simplify the expression.

Step 1: Simplify the Numerator

Using the Power of a Power rule, we can simplify the numerator as follows:

$\left(a^5\right)^3 = a^{5 \cdot 3} = a^{15}$

So, the expression becomes:

$\frac{a^{15}}{a^7}$

Step 2: Simplify the Expression

Now that we have simplified the numerator, we can use the Quotient of Powers rule to simplify the expression. Subtracting the exponents, we get:

$\frac{a^{15}}{a^7} = a^{15-7} = a^8$

Therefore, the simplified expression is $a^8$.

Conclusion

In this article, we have used the properties of exponents to simplify the expression $\frac{\left(a^5\right)^3}{a^7}$. We have applied the Power of a Power rule to simplify the numerator and the Quotient of Powers rule to simplify the expression. By following these steps, we have arrived at the simplified expression $a^8$. This example demonstrates the importance of understanding the rules of exponents and how to apply them to simplify complex expressions.

Example Problems

Here are some example problems that demonstrate the use of the properties of exponents:

Example 1

Simplify the expression $\frac{\left(a^3\right)^2}{a^5}$.

Using the Power of a Power rule, we can simplify the numerator as follows:

$\left(a^3\right)^2 = a^{3 \cdot 2} = a^6$

So, the expression becomes:

$\frac{a^6}{a^5}$

Using the Quotient of Powers rule, we can simplify the expression as follows:

$\frac{a^6}{a^5} = a^{6-5} = a^1 = a$

Therefore, the simplified expression is $a$.

Example 2

Simplify the expression $\frac{a^7}{\left(a^2\right)^3}$.

Using the Power of a Power rule, we can simplify the denominator as follows:

$\left(a^2\right)^3 = a^{2 \cdot 3} = a^6$

So, the expression becomes:

$\frac{a^7}{a^6}$

Using the Quotient of Powers rule, we can simplify the expression as follows:

$\frac{a^7}{a^6} = a^{7-6} = a^1 = a$

Therefore, the simplified expression is $a$.

Practice Problems

Here are some practice problems that demonstrate the use of the properties of exponents:

Problem 1

Simplify the expression $\frac{\left(a^4\right)^2}{a^9}$.

Problem 2

Simplify the expression $\frac{a^8}{\left(a^3\right)^2}$.

Problem 3

Simplify the expression $\frac{\left(a^2\right)^4}{a^6}$.

Answer Key

Here are the answers to the practice problems:

Problem 1

$\frac{\left(a^4\right)^2}{a^9} = \frac{a^{4 \cdot 2}}{a^9} = \frac{a^8}{a^9} = a^{8-9} = a^{-1}$

Problem 2

$\frac{a^8}{\left(a^3\right)^2} = \frac{a^8}{a^{3 \cdot 2}} = \frac{a^8}{a^6} = a^{8-6} = a^2$

Problem 3

$\frac{\left(a^2\right)^4}{a^6} = \frac{a^{2 \cdot 4}}{a^6} = \frac{a^8}{a^6} = a^{8-6} = a^2$

Conclusion

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Q: What are the properties of exponents?

A: The properties of exponents are a set of rules that govern how to simplify and manipulate exponential expressions. The three main properties of exponents are:

• Product of Powers: When multiplying two powers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers: When dividing two powers with the same base, subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify an exponential expression using the properties of exponents?

A: To simplify an exponential expression using the properties of exponents, follow these steps:

1. Identify the base and the exponents in the expression.
2. Apply the Product of Powers rule to simplify the expression if it involves multiplying two powers with the same base.
3. Apply the Power of a Power rule to simplify the expression if it involves raising a power to another power.
4. Apply the Quotient of Powers rule to simplify the expression if it involves dividing two powers with the same base.
5. Simplify the resulting expression by combining like terms.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a base to a certain exponent. For example, $a^m$ is a power, where $a$ is the base and $m$ is the exponent. An exponent, on the other hand, is the number that is raised to a power. For example, $m$ is an exponent in the expression $a^m$.

Q: Can I simplify an exponential expression with a negative exponent?

A: Yes, you can simplify an exponential expression with a negative exponent. When simplifying an expression with a negative exponent, you can rewrite it as a fraction with a positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, follow these steps:

1. Identify the base and the exponents in the expression.
2. Apply the Product of Powers rule to simplify the expression if it involves multiplying two powers with the same base.
3. Apply the Power of a Power rule to simplify the expression if it involves raising a power to another power.
4. Apply the Quotient of Powers rule to simplify the expression if it involves dividing two powers with the same base.
5. Simplify the resulting expression by combining like terms.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent. When simplifying an expression with a zero exponent, the result is always 1. For example, $a^0 = 1$.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, follow these steps:

1. Identify the base and the exponents in the expression.
2. Apply the Power of a Power rule to simplify the expression if it involves raising a power to another power.
3. Simplify the resulting expression by combining like terms.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponential expressions with the properties of exponents. We have covered topics such as the properties of exponents, simplifying expressions with multiple exponents, and simplifying expressions with negative, zero, and fractional exponents. By following the steps outlined in this article, you should be able to simplify complex exponential expressions with ease.

Practice Problems

Here are some practice problems that demonstrate the use of the properties of exponents:

Problem 1

Simplify the expression $\frac{\left(a^3\right)^2}{a^5}$.

Problem 2

Simplify the expression $\frac{a^8}{\left(a^3\right)^2}$.

Problem 3

Simplify the expression $\frac{\left(a^2\right)^4}{a^6}$.

Answer Key

Here are the answers to the practice problems:

Problem 1

$\frac{\left(a^3\right)^2}{a^5} = \frac{a^{3 \cdot 2}}{a^5} = \frac{a^6}{a^5} = a^{6-5} = a^1 = a$

Problem 2

$\frac{a^8}{\left(a^3\right)^2} = \frac{a^8}{a^{3 \cdot 2}} = \frac{a^8}{a^6} = a^{8-6} = a^2$

Problem 3

$\frac{\left(a^2\right)^4}{a^6} = \frac{a^{2 \cdot 4}}{a^6} = \frac{a^8}{a^6} = a^{8-6} = a^2$