Simplify The Expression.\[$(8t)^4\$\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. It involves rewriting complex expressions in a simpler form, making it easier to understand and work with. In this article, we will simplify the expression $(8t)^4$ using the rules of exponents.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^3$ means $a \times a \times a$. When we have an exponent raised to another exponent, we can simplify it using the rule of exponents. The rule states that $(a^m)^n = a^{m \times n}$.

Simplifying the Expression

To simplify the expression $(8t)^4$, we can use the rule of exponents. We will multiply the exponent 4 by the exponent of the base, which is 1 (since $8t$ is already in its simplest form).

$(8t)^4 = (8t)^{1 \times 4}$ $(8t)^4 = 8^4 \times t^4$

Evaluating the Exponents

Now that we have simplified the expression, we need to evaluate the exponents. $8^4$ means $8 \times 8 \times 8 \times 8$, and $t^4$ means $t \times t \times t \times t$.

$8^4 = 8 \times 8 \times 8 \times 8 = 4096$ $t^4 = t \times t \times t \times t = t^4$

Combining the Results

Now that we have evaluated the exponents, we can combine the results to get the final simplified expression.

$(8t)^4 = 8^4 \times t^4$ $(8t)^4 = 4096 \times t^4$ $(8t)^4 = 4096t^4$

Conclusion

In this article, we simplified the expression $(8t)^4$ using the rules of exponents. We used the rule $(a^m)^n = a^{m \times n}$ to simplify the expression, and then evaluated the exponents to get the final result. The simplified expression is $4096t^4$.

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in physics, we use exponents to describe the motion of objects. In economics, we use exponents to describe the growth of populations. In computer science, we use exponents to describe the complexity of algorithms.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Use the rule $(a^m)^n = a^{m \times n}$ to simplify expressions with exponents.
• Evaluate the exponents by multiplying the base by itself as many times as the exponent.
• Combine the results to get the final simplified expression.

Practice Problems

Here are some practice problems to help you practice simplifying expressions:

• Simplify the expression $(3x)^5$
• Simplify the expression $(2y)^3$
• Simplify the expression $(4z)^2$

Answer Key

Here are the answers to the practice problems:

• $(3x)^5 = 3^5 \times x^5 = 243x^5$
• $(2y)^3 = 2^3 \times y^3 = 8y^3$
• $(4z)^2 = 4^2 \times z^2 = 16z^2$

Conclusion

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Introduction

In our previous article, we simplified the expression $(8t)^4$ using the rules of exponents. We used the rule $(a^m)^n = a^{m \times n}$ to simplify the expression, and then evaluated the exponents to get the final result. The simplified expression is $4096t^4$. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the rule for simplifying expressions with exponents?

A: The rule for simplifying expressions with exponents is $(a^m)^n = a^{m \times n}$. This means that when we have an exponent raised to another exponent, we can simplify it by multiplying the two exponents.

Q: How do I evaluate the exponents in a simplified expression?

A: To evaluate the exponents in a simplified expression, we need to multiply the base by itself as many times as the exponent. For example, if we have the expression $8^4$, we need to multiply 8 by itself 4 times: $8 \times 8 \times 8 \times 8 = 4096$.

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

A: A base is the number or variable that is being raised to a power, while an exponent is the power to which the base is being raised. For example, in the expression $8^4$, 8 is the base and 4 is the exponent.

Q: Can I simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents. To do this, we need to use the rule $a^{-n} = \frac{1}{a^n}$. This means that a negative exponent is equal to the reciprocal of the positive exponent.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we need to use the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. This means that a fractional exponent is equal to the nth root of the base raised to the power of the numerator.

Q: Can I simplify expressions with variables as exponents?

A: Yes, we can simplify expressions with variables as exponents. To do this, we need to use the rule $(a^m)^n = a^{m \times n}$. This means that we can simplify expressions with variables as exponents by multiplying the exponents.

Q: How do I simplify expressions with multiple bases and exponents?

A: To simplify expressions with multiple bases and exponents, we need to use the rule $(a^m)^n = a^{m \times n}$. This means that we can simplify expressions with multiple bases and exponents by multiplying the exponents.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions. We discussed the rule for simplifying expressions with exponents, how to evaluate the exponents in a simplified expression, and how to simplify expressions with negative, fractional, and variable exponents. We also discussed how to simplify expressions with multiple bases and exponents. By following these rules and tips, you can simplify expressions with ease and become a master of algebra.

Practice Problems

Here are some practice problems to help you practice simplifying expressions:

• Simplify the expression $(3x)^5$
• Simplify the expression $(2y)^3$
• Simplify the expression $(4z)^2$
• Simplify the expression $(8t)^4$
• Simplify the expression $(x^2)^3$

Answer Key

Here are the answers to the practice problems:

• $(3x)^5 = 3^5 \times x^5 = 243x^5$
• $(2y)^3 = 2^3 \times y^3 = 8y^3$
• $(4z)^2 = 4^2 \times z^2 = 16z^2$
• $(8t)^4 = 8^4 \times t^4 = 4096t^4$
• $(x^2)^3 = x^{2 \times 3} = x^6$

Conclusion

In this article, we simplified the expression $(8t)^4$ using the rules of exponents. We also answered some frequently asked questions about simplifying expressions. By following these rules and tips, you can simplify expressions with ease and become a master of algebra.