Understanding Exponents and Equivalents

Oct 13, 2025 by magicalceylontours.com 128 views

**Which Choice is Equivalent to the Expression Below?**

Understanding Exponents and Equivalents

Exponents are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will explore the concept of exponents and how to find equivalent expressions.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $5^3$ can be read as "5 to the power of 3" and is equivalent to $5 \times 5 \times 5$ . Exponents can be positive, negative, or fractional, and they can be used to represent very large or very small numbers.

Understanding the Given Expression

The given expression is $5^{9.969}$ . This expression represents a number that is the result of raising 5 to the power of 9.969. To simplify this expression, we need to understand the properties of exponents.

Properties of Exponents

There are several properties of exponents that we can use to simplify the given expression. These properties include:

Product of Powers: When we multiply two numbers with the same base, we can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$ .
Power of a Power: When we raise a number with an exponent to another power, we can multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .
Quotient of Powers: When we divide two numbers with the same base, we can subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$ .

Simplifying the Given Expression

Using the properties of exponents, we can simplify the given expression as follows:

$5^{9.969} = 5^9 \cdot 5^{0.969}$

We can further simplify this expression by using the property of product of powers:

$5^9 \cdot 5^{0.969} = 5^{9 + 0.969}$

Now, we can use the property of power of a power to simplify the exponent:

$5^{9 + 0.969} = 5^9 \cdot 5^{0.969}$

However, we can simplify this expression further by using the property of product of powers:

$5^9 \cdot 5^{0.969} = 5^{9 + 0.969}$

Now, we can simplify the exponent by adding the numbers:

$9 + 0.969 = 9.969$

Therefore, the simplified expression is:

$5^{9.969} = 5^{9.969}$

Which Choice is Equivalent to the Expression Below?

Now that we have simplified the given expression, we can compare it to the choices provided. The choices are:

A. $5^9 + 5^{9 / 10} + 6^{6 / 100}$ B. $5^9 \cdot 5^{9 / 10} \cdot 5^{6 / 100} \cdot 5^{9 / 1000}$ C. $5^{9 + 9 / 10 + 9 / 10 + 6 / 1000}$ D. $5^9 \cdot 5^{9 / 10} \cdot 5^{6 / 100} \cdot 5^{9 / 1000}$

Answer

The correct answer is C. $5^{9 + 9 / 10 + 9 / 10 + 6 / 1000}$ .

Explanation

The correct answer is C because it is the only choice that is equivalent to the simplified expression $5^{9.969}$ . The other choices are not equivalent to the simplified expression.

Conclusion

In this article, we have explored the concept of exponents and how to find equivalent expressions. We have simplified the given expression $5^{9.969}$ using the properties of exponents and compared it to the choices provided. The correct answer is C. $5^{9 + 9 / 10 + 9 / 10 + 6 / 1000}$ .

Frequently Asked Questions

Q: What is the definition of an exponent? A: An exponent is a shorthand way of representing repeated multiplication.

Q: What are the properties of exponents? A: The properties of exponents include the product of powers, power of a power, and quotient of powers.

Q: How do I simplify an expression with exponents? A: To simplify an expression with exponents, you can use the properties of exponents, such as the product of powers, power of a power, and quotient of powers.

Q: What is the correct answer to the given problem? A: The correct answer is C. $5^{9 + 9 / 10 + 9 / 10 + 6 / 1000}$ .

Q: Why is the correct answer C? A: The correct answer is C because it is the only choice that is equivalent to the simplified expression $5^{9.969}$ .