Express The Following Using A Positive Exponent, Then Simplify The Expression.$3^{-3}$Write Using A Positive Exponent. Do Not Evaluate.$3^{-3} = \square$

**Expressing Negative Exponents: A Comprehensive Guide** =====================================================

What is a Negative Exponent?

A negative exponent is a mathematical expression that represents a number raised to a power that is less than zero. In other words, it is the reciprocal of a positive exponent. For example, $3^{-3}$ can be read as "3 to the power of negative 3" or "the reciprocal of 3 to the power of 3".

Expressing Negative Exponents using Positive Exponents

To express a negative exponent using a positive exponent, we can use the following rule:

$a^{-n} = \frac{1}{a^n}$

where $a$ is a non-zero number and $n$ is a positive integer.

Example: Expressing $3^{-3}$ using a Positive Exponent

Using the rule above, we can express $3^{-3}$ as:

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

Simplifying the Expression

To simplify the expression, we can evaluate the exponent:

$3^3 = 3 \times 3 \times 3 = 27$

So, the simplified expression is:

$3^{-3} = \frac{1}{27}$

Q&A: Expressing Negative Exponents

Q: What is the rule for expressing negative exponents using positive exponents?

A: The rule is $a^{-n} = \frac{1}{a^n}$, where $a$ is a non-zero number and $n$ is a positive integer.

Q: How do I express $2^{-4}$ using a positive exponent?

A: Using the rule above, we can express $2^{-4}$ as:

$2^{-4} = \frac{1}{2^4}$

Q: What is the simplified expression for $2^{-4}$?

A: To simplify the expression, we can evaluate the exponent:

$2^4 = 2 \times 2 \times 2 \times 2 = 16$

So, the simplified expression is:

$2^{-4} = \frac{1}{16}$

Q: Can I express a negative exponent using a positive exponent if the base is zero?

A: No, you cannot express a negative exponent using a positive exponent if the base is zero. The rule only applies to non-zero bases.

Q: How do I express $x^{-2}$ using a positive exponent?

A: Using the rule above, we can express $x^{-2}$ as:

$x^{-2} = \frac{1}{x^2}$

Q: What is the simplified expression for $x^{-2}$?

A: To simplify the expression, we can leave it as is, since $x^2$ is already a positive exponent.

Q: Can I express a negative exponent using a positive exponent if the exponent is a fraction?

A: Yes, you can express a negative exponent using a positive exponent if the exponent is a fraction. For example:

$x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}}$

Q: How do I express $y^{-\frac{1}{3}}$ using a positive exponent?

A: Using the rule above, we can express $y^{-\frac{1}{3}}$ as:

$y^{-\frac{1}{3}} = \frac{1}{y^{\frac{1}{3}}}$

Q: What is the simplified expression for $y^{-\frac{1}{3}}$?

A: To simplify the expression, we can leave it as is, since $y^{\frac{1}{3}}$ is already a positive exponent.

Conclusion

Expressing negative exponents using positive exponents is a useful technique in mathematics. By using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify expressions and make them easier to work with. Remember to always evaluate the exponent and simplify the expression to get the final answer.