Consider The Expression $6 \cdot 6^0 \cdot 6^{-3}$.Which Statements Are True About The Expression? Check All That Apply.- The 6 Without An Exponent Is Equivalent To The 6 Having A 0 Exponent.- The Sum Of The Exponents Is 2.- Multiply The

When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore the expression $6 \cdot 6^0 \cdot 6^{-3}$ and examine the statements that are true about it.

The Basics of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $6^3$ means $6 \cdot 6 \cdot 6$. The exponent tells us how many times to multiply the base number. In this case, the base number is 6, and the exponent is 3.

The Zero Exponent Rule

One of the fundamental rules of exponents is the zero exponent rule. This rule states that any number raised to the power of 0 is equal to 1. In other words, $a^0 = 1$, where $a$ is any non-zero number. This rule applies to the expression $6^0$, which is equal to 1.

The Negative Exponent Rule

Another important rule of exponents is the negative exponent rule. This rule states that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power. In other words, $a^{-n} = \frac{1}{a^n}$, where $a$ is any non-zero number and $n$ is a positive integer. This rule applies to the expression $6^{-3}$, which is equal to $\frac{1}{6^3}$.

Evaluating the Expression

Now that we have a good understanding of the rules of exponents, let's evaluate the expression $6 \cdot 6^0 \cdot 6^{-3}$. Using the zero exponent rule, we know that $6^0 = 1$. Using the negative exponent rule, we know that $6^{-3} = \frac{1}{6^3}$. Therefore, the expression can be simplified as follows:

$$6 \cdot 6^0 \cdot 6^{-3} = 6 \cdot 1 \cdot \frac{1}{6^3} = \frac{6}{6^3} = \frac{1}{6^2} = \frac{1}{36}$$

Analyzing the Statements

Now that we have evaluated the expression, let's analyze the statements that were made about it.

• The 6 without an exponent is equivalent to the 6 having a 0 exponent. This statement is true. As we discussed earlier, the zero exponent rule states that any number raised to the power of 0 is equal to 1. Therefore, the 6 without an exponent is indeed equivalent to the 6 having a 0 exponent.
• The sum of the exponents is 2. This statement is false. The sum of the exponents is actually 0 + (-3) = -3, not 2.
• Multiply the This statement is incomplete and does not make sense in the context of the expression.

Conclusion

In conclusion, the expression $6 \cdot 6^0 \cdot 6^{-3}$ can be simplified to $\frac{1}{36}$. The statements that are true about the expression are that the 6 without an exponent is equivalent to the 6 having a 0 exponent, and the expression can be simplified using the rules of exponents.

Key Takeaways

• The zero exponent rule states that any number raised to the power of 0 is equal to 1.
• The negative exponent rule states that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power.
• The expression $6 \cdot 6^0 \cdot 6^{-3}$ can be simplified to $\frac{1}{36}$ using the rules of exponents.

Further Reading

If you're interested in learning more about exponents and their applications, I recommend checking out the following resources:

• Khan Academy's Exponents course
• Mathway's Exponents tutorial
• Wolfram MathWorld's Exponents page

In the previous article, we explored the expression $6 \cdot 6^0 \cdot 6^{-3}$ and examined the statements that are true about it. In this article, we will answer some frequently asked questions about exponents and provide additional examples to help solidify your understanding.

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

A: A base is the number being raised to a power, while an exponent is the power to which the base is being raised. For example, in the expression $6^3$, the base is 6 and the exponent is 3.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of 0 is equal to 1. In other words, $a^0 = 1$, where $a$ is any non-zero number.

Q: What is the negative exponent rule?

A: The negative exponent rule states that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power. In other words, $a^{-n} = \frac{1}{a^n}$, where $a$ is any non-zero number and $n$ is a positive integer.

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

A: To simplify an expression with multiple exponents, you can use the rules of exponents to combine the exponents. For example, in the expression $6^2 \cdot 6^3$, you can combine the exponents by adding them together: $6^{2+3} = 6^5$.

Q: What is the difference between $a^m \cdot a^n$ and $a^{m+n}$?

A: $a^m \cdot a^n$ is the product of two numbers raised to different powers, while $a^{m+n}$ is the result of combining the exponents by adding them together. For example, in the expression $2^3 \cdot 2^4$, you can combine the exponents by adding them together: $2^{3+4} = 2^7$. However, if you multiply the numbers first, you get $2^3 \cdot 2^4 = 2^7$, which is the same result.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by using the negative exponent rule. For example, in the expression $6^{-3}$, you can simplify it by taking the reciprocal of $6^3$: $\frac{1}{6^3}$.

Q: How do I evaluate an expression with a zero exponent?

A: To evaluate an expression with a zero exponent, you can use the zero exponent rule, which states that any number raised to the power of 0 is equal to 1. For example, in the expression $6^0$, you can simplify it by using the zero exponent rule: $6^0 = 1$.

Q: Can I simplify an expression with multiple zero exponents?

A: Yes, you can simplify an expression with multiple zero exponents by using the zero exponent rule. For example, in the expression $6^0 \cdot 6^0$, you can simplify it by using the zero exponent rule: $6^0 \cdot 6^0 = 1 \cdot 1 = 1$.

Q: How do I evaluate an expression with a negative exponent and a zero exponent?

A: To evaluate an expression with a negative exponent and a zero exponent, you can use the rules of exponents to simplify the expression. For example, in the expression $6^{-3} \cdot 6^0$, you can simplify it by using the negative exponent rule and the zero exponent rule: $6^{-3} \cdot 6^0 = \frac{1}{6^3} \cdot 1 = \frac{1}{6^3}$.

Conclusion

In conclusion, exponents are a powerful tool for simplifying complex expressions and solving mathematical problems. By understanding the rules of exponents and how to apply them, you can simplify expressions with multiple exponents, negative exponents, and zero exponents. Remember to use the zero exponent rule, the negative exponent rule, and the rules for combining exponents to simplify expressions and solve problems.

Key Takeaways

• The zero exponent rule states that any number raised to the power of 0 is equal to 1.
• The negative exponent rule states that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power.
• The rules for combining exponents state that when multiplying numbers with the same base, you can add the exponents together.
• When simplifying an expression with multiple exponents, you can use the rules of exponents to combine the exponents.

Further Reading

If you're interested in learning more about exponents and their applications, I recommend checking out the following resources:

• Khan Academy's Exponents course
• Mathway's Exponents tutorial
• Wolfram MathWorld's Exponents page

By understanding the rules of exponents and how to apply them, you can simplify complex expressions and solve a wide range of mathematical problems.