Evaluating the Terms with Exponents

Oct 13, 2025 by windailys.com 27 views

When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we'll 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.

Zero Exponent

A zero exponent is a special case that can be a bit tricky to understand. When a number has a zero exponent, it is equal to 1. This is a fundamental property of exponents that is essential to remember.

The Expression $6 \cdot 6^0 \cdot 6^{-3}$

Now, let's examine the expression $6 \cdot 6^0 \cdot 6^{-3}$ . We can start by evaluating the terms with exponents.

Evaluating the Terms with Exponents

When we have multiple terms with exponents, we can simplify them by applying the rules of exponents. In this case, we have $6^0$ and $6^{-3}$ .

$6^0$ is equal to 1, since any number raised to the power of 0 is equal to 1.
$6^{-3}$ is equal to $\frac{1}{6^3}$ , since a negative exponent means we are taking the reciprocal of the base number raised to the positive exponent.

Simplifying the Expression

Now that we have evaluated the terms with exponents, we can simplify the expression.

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

We can simplify this expression further by canceling out the common factors.

$6 \cdot 1 \cdot \frac{1}{6^3} = \frac{6}{6^3}$

Simplifying the Fraction

We can simplify the fraction by applying the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ .

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

Evaluating the Final Expression

Now that we have simplified the expression, we can evaluate it.

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

Evaluating the Statements

Now that we have evaluated the expression, we can examine 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, since $6^0$ is equal to 1.
The sum of the exponents is 2. This statement is false, since the sum of the exponents is actually -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 that the sum of the exponents is -2.

Key Takeaways

Exponents are a shorthand way of representing repeated multiplication.
A zero exponent is equal to 1.
A negative exponent means we are taking the reciprocal of the base number raised to the positive exponent.
We can simplify expressions with exponents by applying the rules of exponents.
The sum of the exponents in an expression is not necessarily equal to the exponent of the final result.

Final Thoughts

In this article, we'll answer some of the most frequently asked questions about exponents.

Q: What is an exponent?

A: An exponent is a shorthand way of representing repeated multiplication. For example, $6^3$ means $6 \cdot 6 \cdot 6$ .

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, we add the exponents. For example, $6^2 \cdot 6^3 = 6^{2+3} = 6^5$ .

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, we subtract the exponents. For example, $\frac{6^2}{6^3} = 6^{2-3} = 6^{-1}$ .

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, we multiply the exponents. For example, $(6^2)^3 = 6^{2 \cdot 3} = 6^6$ .

Q: What is the rule for evaluating an expression with a zero exponent?

A: When evaluating an expression with a zero exponent, we know that the result is always 1. For example, $6^0 = 1$ .

Q: What is the rule for evaluating an expression with a negative exponent?

A: When evaluating an expression with a negative exponent, we know that the result is the reciprocal of the base number raised to the positive exponent. For example, $6^{-3} = \frac{1}{6^3}$ .

Q: Can you give an example of how to simplify an expression with multiple exponents?

A: Yes, here's an example:

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

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

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

Q: Can you give an example of how to evaluate an expression with multiple bases and exponents?

A: Yes, here's an example:

$(2^3 \cdot 3^2)^4 = (2^3)^4 \cdot (3^2)^4 = 2^{3 \cdot 4} \cdot 3^{2 \cdot 4} = 2^{12} \cdot 3^{8}$

Q: What is the rule for evaluating an expression with a fractional exponent?

A: When evaluating an expression with a fractional exponent, we know that the result is the nth root of the base number raised to the power of m. For example, $6^{\frac{1}{2}} = \sqrt{6}$ .

Q: Can you give an example of how to simplify an expression with a fractional exponent?

A: Yes, here's an example:

$6^{\frac{1}{2}} = \sqrt{6} = \frac{\sqrt{6}}{\sqrt{6}} \cdot \sqrt{6} = \frac{6^{\frac{1}{2}}}{6^{\frac{1}{2}}} \cdot 6^{\frac{1}{2}} = 6^{\frac{1}{2}}$

Conclusion

In conclusion, exponents are a fundamental concept in mathematics that can be used to simplify complex expressions and solve problems with ease. By mastering the rules and properties of exponents, we can evaluate expressions with confidence and accuracy.

Key Takeaways

Exponents are a shorthand way of representing repeated multiplication.
The rule for multiplying exponents with the same base is to add the exponents.
The rule for dividing exponents with the same base is to subtract the exponents.
The rule for raising a power to a power is to multiply the exponents.
The rule for evaluating an expression with a zero exponent is that the result is always 1.
The rule for evaluating an expression with a negative exponent is that the result is the reciprocal of the base number raised to the positive exponent.

Final Thoughts

Understanding exponents and their applications is essential for success in mathematics. By mastering the rules and properties of exponents, we can simplify complex expressions and solve problems with ease. Whether you're a student or a professional, having a solid grasp of exponents is crucial for achieving your goals.