Write $(4t)^5$ Without Exponents.

Understanding Exponents and Their Representation

Exponents are a shorthand way of representing repeated multiplication of a number. In the expression $(4t)^5$, the exponent 5 indicates that the base $4t$ should be multiplied by itself 5 times. This can be represented as $(4t) \times (4t) \times (4t) \times (4t) \times (4t)$. In this article, we will explore how to write this expression without using exponents.

The Concept of Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. In the case of $(4t)^5$, the exponent 5 represents the power to which the base $4t$ is raised. This can be thought of as multiplying the base by itself 5 times, as mentioned earlier.

Expanding the Expression without Exponents

To write $(4t)^5$ without exponents, we need to multiply the base $4t$ by itself 5 times. This can be done using the distributive property of multiplication over addition. The distributive property states that for any numbers a, b, and c, the following equation holds:

$$a(b + c) = ab + ac$$

We can use this property to expand the expression $(4t)^5$ as follows:

$$(4t)^5 = (4t) \times (4t) \times (4t) \times (4t) \times (4t)$$

Using the distributive property, we can rewrite this expression as:

$$(4t)^5 = 4t \times 4t \times 4t \times 4t \times 4t$$

Now, we can simplify each multiplication step:

$$(4t)^5 = (4 \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4)$$

Simplifying further, we get:

$$(4t)^5 = 16 \times 4t \times 4t \times 4t \times 4t$$

Continuing to simplify, we get:

$$(4t)^5 = 16 \times 4^4t^5$$

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

$$(4t)^5 = 16 \times 4^4t^5 = 16 \times 4^4 \times t^5$$

Now, we can simplify the exponent $4^4$ as follows:

$$4^4 = 4 \times 4 \times 4 \times 4 = 256$$

Substituting this value back into the expression, we get:

$$(4t)^5 = 16 \times 256 \times t^5$$

Simplifying further, we get:

$$(4t)^5 = 4096t^5$$

Conclusion

In this article, we have explored how to write the expression $(4t)^5$ without using exponents. We used the distributive property of multiplication over addition to expand the expression, and then simplified each multiplication step to arrive at the final answer. The result is $(4t)^5 = 4096t^5$.

Frequently Asked Questions

• What is the exponent 5 in the expression $(4t)^5$?
  • The exponent 5 represents the power to which the base $4t$ is raised.
• How do you expand the expression $(4t)^5$ without using exponents?
  • You can use the distributive property of multiplication over addition to expand the expression.
• What is the final answer to the expression $(4t)^5$?
  • The final answer is $(4t)^5 = 4096t^5$.

Final Thoughts

In conclusion, writing the expression $(4t)^5$ without exponents requires a clear understanding of the concept of exponentiation and the distributive property of multiplication over addition. By following the steps outlined in this article, you can arrive at the final answer of $(4t)^5 = 4096t^5$.

Understanding Exponents and Their Representation

Exponents are a shorthand way of representing repeated multiplication of a number. In the expression $(4t)^5$, the exponent 5 indicates that the base $4t$ should be multiplied by itself 5 times. This can be represented as $(4t) \times (4t) \times (4t) \times (4t) \times (4t)$. In this article, we will explore how to write this expression without using exponents.

The Concept of Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. In the case of $(4t)^5$, the exponent 5 represents the power to which the base $4t$ is raised. This can be thought of as multiplying the base by itself 5 times, as mentioned earlier.

Expanding the Expression without Exponents

To write $(4t)^5$ without exponents, we need to multiply the base $4t$ by itself 5 times. This can be done using the distributive property of multiplication over addition. The distributive property states that for any numbers a, b, and c, the following equation holds:

$$a(b + c) = ab + ac$$

We can use this property to expand the expression $(4t)^5$ as follows:

$$(4t)^5 = (4t) \times (4t) \times (4t) \times (4t) \times (4t)$$

Using the distributive property, we can rewrite this expression as:

$$(4t)^5 = 4t \times 4t \times 4t \times 4t \times 4t$$

Now, we can simplify each multiplication step:

$$(4t)^5 = (4 \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4)$$

Simplifying further, we get:

$$(4t)^5 = 16 \times 4t \times 4t \times 4t \times 4t$$

Continuing to simplify, we get:

$$(4t)^5 = 16 \times 4^4t^5$$

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

$$(4t)^5 = 16 \times 4^4t^5 = 16 \times 4^4 \times t^5$$

Now, we can simplify the exponent $4^4$ as follows:

$$4^4 = 4 \times 4 \times 4 \times 4 = 256$$

Substituting this value back into the expression, we get:

$$(4t)^5 = 16 \times 256 \times t^5$$

Simplifying further, we get:

$$(4t)^5 = 4096t^5$$

Q&A

Q: What is the exponent 5 in the expression $(4t)^5$?

A: The exponent 5 represents the power to which the base $4t$ is raised.

Q: How do you expand the expression $(4t)^5$ without using exponents?

A: You can use the distributive property of multiplication over addition to expand the expression.

Q: What is the final answer to the expression $(4t)^5$?

A: The final answer is $(4t)^5 = 4096t^5$.

Q: Can you explain the concept of exponentiation?

A: Exponentiation is a mathematical operation that involves raising a number to a power. In the case of $(4t)^5$, the exponent 5 represents the power to which the base $4t$ is raised.

Q: How do you simplify the expression $(4t)^5$?

A: You can simplify the expression by multiplying the base $4t$ by itself 5 times, and then simplifying each multiplication step.

Q: What is the property of exponents that states $a^m \times a^n = a^{m+n}$?

A: This property states that when you multiply two numbers with the same base, you can add their exponents.

Q: Can you provide an example of how to use the distributive property to expand the expression $(4t)^5$?

A: Yes, here is an example:

$$(4t)^5 = (4t) \times (4t) \times (4t) \times (4t) \times (4t)$$

Using the distributive property, we can rewrite this expression as:

$$(4t)^5 = 4t \times 4t \times 4t \times 4t \times 4t$$

Now, we can simplify each multiplication step:

$$(4t)^5 = (4 \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4) \times (t \times 4)$$

Simplifying further, we get:

$$(4t)^5 = 16 \times 4t \times 4t \times 4t \times 4t$$

Continuing to simplify, we get:

$$(4t)^5 = 16 \times 4^4t^5$$

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

$$(4t)^5 = 16 \times 4^4t^5 = 16 \times 4^4 \times t^5$$

Now, we can simplify the exponent $4^4$ as follows:

$$4^4 = 4 \times 4 \times 4 \times 4 = 256$$

Substituting this value back into the expression, we get:

$$(4t)^5 = 16 \times 256 \times t^5$$

Simplifying further, we get:

$$(4t)^5 = 4096t^5$$

Conclusion

In this article, we have explored how to write the expression $(4t)^5$ without using exponents. We used the distributive property of multiplication over addition to expand the expression, and then simplified each multiplication step to arrive at the final answer. The result is $(4t)^5 = 4096t^5$.

Final Thoughts

In conclusion, writing the expression $(4t)^5$ without exponents requires a clear understanding of the concept of exponentiation and the distributive property of multiplication over addition. By following the steps outlined in this article, you can arrive at the final answer of $(4t)^5 = 4096t^5$.