A) Simplify The Expression: 7 − 3 × 3 4 3 − 2 × 7 5 × 5 − 5 = \frac{7^{-3} \times 3^4}{3^{-2} \times 7^5 \times 5^{-5}} = 3 − 2 × 7 5 × 5 − 5 7 − 3 × 3 4 =

Mar 8, 2025 by ADMIN 158 views

**Simplify the Expression: A Comprehensive Guide**

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying a given expression involving exponents and fractions. We will break down the expression step by step, using the properties of exponents and fractions to simplify it.

Understanding Exponents and Fractions

Before we dive into the simplification process, let's review the basics of exponents and fractions.

Exponents: Exponents are a shorthand way of representing repeated multiplication. For example, $a^3$ means $a \times a \times a$ . Exponents can also be negative, which means we are dealing with fractions. For example, $a^{-3}$ means $\frac{1}{a^3}$ .
Fractions: Fractions are a way of representing part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). For example, $\frac{1}{2}$ means one half.

The Given Expression

The given expression is:

\frac{7^{-3} \times 3^4}{3^{-2} \times 7^5 \times 5^{-5}}

Our goal is to simplify this expression using the properties of exponents and fractions.

Step 1: Simplify the Numerator

Let's start by simplifying the numerator:

7^{-3} \times 3^4

Using the property of exponents that states $a^m \times a^n = a^{m+n}$ , we can rewrite the numerator as:

7^{-3} \times 3^4 = 7^{-3} \times 3^4 = \frac{3^4}{7^3}

Step 2: Simplify the Denominator

Next, let's simplify the denominator:

3^{-2} \times 7^5 \times 5^{-5}

Using the property of exponents that states $a^m \times a^n = a^{m+n}$ , we can rewrite the denominator as:

3^{-2} \times 7^5 \times 5^{-5} = \frac{7^5}{3^2} \times \frac{1}{5^5}

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can rewrite the original expression as:

\frac{\frac{3^4}{7^3}}{\frac{7^5}{3^2} \times \frac{1}{5^5}}

Using the property of fractions that states $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ , we can rewrite the expression as:

\frac{3^4}{7^3} \times \frac{5^5}{3^2 \times 7^5}

Step 4: Simplify the Expression Further

Using the property of exponents that states $a^m \times a^n = a^{m+n}$ , we can rewrite the expression as:

\frac{3^4 \times 5^5}{7^3 \times 3^2 \times 7^5}

Using the property of exponents that states $a^m \times a^n = a^{m+n}$ , we can rewrite the expression as:

\frac{3^2 \times 5^5}{7^8}

Conclusion

In this article, we simplified the given expression using the properties of exponents and fractions. We broke down the expression step by step, using the properties of exponents and fractions to simplify it. The final simplified expression is:

\frac{3^2 \times 5^5}{7^8}

This expression cannot be simplified further using the properties of exponents and fractions.

Final Answer

The final answer is:

\boxed{\frac{9 \times 3125}{5764801}}

Note

The final answer can be simplified further using decimal arithmetic. However, we will not do that here as we are only interested in the exact value of the expression.

References

[1] "Exponents and Fractions" by Khan Academy
[2] "Simplifying Expressions" by Mathway

Additional Resources

[1] "Exponents and Fractions" by IXL
[2] "Simplifying Expressions" by Purplemath
Simplify the Expression: A Comprehensive Guide =====================================================

Q&A: Simplifying Expressions

In the previous article, we simplified the given expression using the properties of exponents and fractions. However, we received many questions from readers who wanted to know more about simplifying expressions. In this article, we will answer some of the most frequently asked questions about simplifying expressions.

Q: What are the properties of exponents?

A: The properties of exponents are a set of rules that help us simplify expressions involving exponents. Some of the most important properties of exponents include:

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 expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to use the properties of exponents. Here are the steps to follow:

Identify the exponents in the expression.
Use the product of powers property to combine the exponents.
Simplify the resulting expression.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of representing part of a whole, while a decimal is a way of representing a fraction as a decimal number. For example, $\frac{1}{2}$ is a fraction, while 0.5 is a decimal.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, we need to divide the numerator by the denominator. For example, to convert $\frac{1}{2}$ to a decimal, we would divide 1 by 2, which gives us 0.5.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number. For example, $a^3$ is a positive exponent, while $a^{-3}$ is a negative exponent.

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

A: To simplify an expression with negative exponents, we need to use the property of exponents that states $a^{-m} = \frac{1}{a^m}$ . Here are the steps to follow:

Identify the negative exponents in the expression.
Use the property of exponents to rewrite the negative exponents as positive exponents.
Simplify the resulting expression.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be simplified to a fraction, while an irrational expression is an expression that cannot be simplified to a fraction. For example, $\frac{1}{2}$ is a rational expression, while $\sqrt{2}$ is an irrational expression.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we need to use the properties of fractions. Here are the steps to follow:

Identify the fractions in the expression.
Use the property of fractions that states $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ to simplify the fractions.
Simplify the resulting expression.