Select The Correct Answer.Which Of The Following Is Equal To The Expression Below? $(100 \cdot 213)^t$A. 96 B. $6 \sqrt[5]{5}$ C. \$5 \sqrt[5]{5}$[/tex\] D. 80

Introduction

Exponential expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of exponents, we can break down even the most daunting expressions into manageable parts. In this article, we will explore how to simplify the expression $(100 \cdot 213)^t$ and determine which of the given options is equal to it.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, in the expression $2^3$, the exponent $3$ tells us to multiply $2$ by itself $3$ times: $2 \cdot 2 \cdot 2 = 8$.

Simplifying the Expression

Now that we have a solid understanding of exponents, let's simplify the expression $(100 \cdot 213)^t$. To do this, we need to apply the property of exponents that states $(ab)^c = a^c \cdot b^c$. In this case, we can rewrite the expression as $(100 \cdot 213)^t = 100^t \cdot 213^t$.

Breaking Down the Expression

Next, we need to break down the expression $100^t \cdot 213^t$ into its prime factors. To do this, we can use the fact that $100 = 2^2 \cdot 5^2$ and $213 = 3 \cdot 71$. Therefore, we can rewrite the expression as $(2^2 \cdot 5^2)^t \cdot (3 \cdot 71)^t$.

Applying the Exponent Property

Now that we have broken down the expression into its prime factors, we can apply the exponent property again. This time, we can rewrite the expression as $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t$.

Simplifying the Expression Further

At this point, we can simplify the expression further by combining like terms. We can rewrite the expression as $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t = (2^t \cdot 5^t)^2 \cdot 3^t \cdot 71^t$.

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options to determine which one is equal to it. Let's take a closer look at each option:

• Option A: 96. This option does not match the simplified expression.
• Option B: $6 \sqrt[5]{5}$. This option does not match the simplified expression.
• Option C: $5 \sqrt[5]{5}$. This option does not match the simplified expression.
• Option D: 80. This option does not match the simplified expression.

Conclusion

In conclusion, the expression $(100 \cdot 213)^t$ cannot be simplified to any of the given options. However, we can rewrite the expression as $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t$. This expression cannot be simplified further without additional information about the value of $t$.

Final Answer

Q: What is the property of exponents that states $(ab)^c = a^c \cdot b^c$?

A: This property is known as the product of powers property. It states that when we have a product of two numbers raised to a power, we can separate the product into two separate powers.

Q: How do I apply the product of powers property to simplify an expression?

A: To apply the product of powers property, we need to identify the numbers that are being multiplied together and raise them to the same power. For example, in the expression $(100 \cdot 213)^t$, we can rewrite it as $100^t \cdot 213^t$.

Q: What is the difference between the product of powers property and the power of a product property?

A: The product of powers property states that $(ab)^c = a^c \cdot b^c$, while the power of a product property states that $(ab)^c = a^c \cdot b^c$. The key difference is that the product of powers property is used when we have a product of two numbers raised to a power, while the power of a product property is used when we have a single number raised to a power.

Q: How do I break down an expression into its prime factors?

A: To break down an expression into its prime factors, we need to identify the prime numbers that multiply together to give the original expression. For example, in the expression $100^t \cdot 213^t$, we can break it down into its prime factors as $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t$.

Q: What is the difference between a prime number and a composite number?

A: A prime number is a number that is divisible only by itself and 1, while a composite number is a number that is divisible by more than just itself and 1. For example, the number 5 is a prime number because it is only divisible by itself and 1, while the number 6 is a composite number because it is divisible by 1, 2, 3, and 6.

Q: How do I simplify an expression further by combining like terms?

A: To simplify an expression further by combining like terms, we need to identify the terms that have the same variable raised to the same power. For example, in the expression $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t$, we can combine the like terms to get $(2^t \cdot 5^t)^2 \cdot 3^t \cdot 71^t$.

Q: What is the final answer to the expression $(100 \cdot 213)^t$?

A: The final answer to the expression $(100 \cdot 213)^t$ is $2^{2t} \cdot 5^{2t} \cdot 3^t \cdot 71^t$. This expression cannot be simplified further without additional information about the value of $t$.

Q: Can I use the product of powers property to simplify an expression with a negative exponent?

A: Yes, you can use the product of powers property to simplify an expression with a negative exponent. For example, in the expression $(100 \cdot 213)^{-t}$, we can rewrite it as $100^{-t} \cdot 213^{-t}$.

Q: How do I apply the product of powers property to simplify an expression with a fractional exponent?

A: To apply the product of powers property to simplify an expression with a fractional exponent, we need to identify the numbers that are being multiplied together and raise them to the same power. For example, in the expression $(100 \cdot 213)^{1/2}$, we can rewrite it as $100^{1/2} \cdot 213^{1/2}$.

Conclusion

In conclusion, simplifying exponential expressions can be a complex and challenging task. However, by applying the product of powers property, breaking down expressions into their prime factors, and combining like terms, we can simplify even the most daunting expressions. Remember to always identify the numbers that are being multiplied together and raise them to the same power, and to use the product of powers property to simplify expressions with negative and fractional exponents.