Applying the Negative Exponent Rule

Oct 13, 2025 by magicalceylontours.com 79 views

Introduction

In mathematics, the concept of exponents is crucial in solving various types of problems. Exponents are used to represent the power or the index of a number. When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will evaluate the expression $\left(9^{\frac{5}{2}}\right)^0 \left(9^{-\frac{3}{2}}\right)$ using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^b$ can be written as $a \times a \times a \times \ldots \times a$ (b times). The exponent $b$ represents the number of times the base $a$ is multiplied by itself.

Evaluating the Expression

To evaluate the expression $\left(9^{\frac{5}{2}}\right)^0 \left(9^{-\frac{3}{2}}\right)$ , we need to apply the properties of exponents. The first property we will use is the zero exponent rule, which states that any non-zero number raised to the power of zero is equal to 1.

$\left(9^{\frac{5}{2}}\right)^0 = 1$

This is because any non-zero number raised to the power of zero is equal to 1.

Applying the Negative Exponent Rule

The second property we will use is the negative exponent rule, which states that $a^{-b} = \frac{1}{a^b}$ . We can apply this rule to the second term in the expression.

$\left(9^{-\frac{3}{2}}\right) = \frac{1}{9^{\frac{3}{2}}}$

Simplifying the Expression

Now that we have applied the properties of exponents, we can simplify the expression.

$\left(9^{\frac{5}{2}}\right)^0 \left(9^{-\frac{3}{2}}\right) = 1 \times \frac{1}{9^{\frac{3}{2}}}$

Evaluating the Expression Further

To evaluate the expression further, we need to simplify the fraction.

$\frac{1}{9^{\frac{3}{2}}} = \frac{1}{(9^{\frac{1}{2}})^3}$

Simplifying the Fraction

We can simplify the fraction by evaluating the expression inside the parentheses.

$9^{\frac{1}{2}} = \sqrt{9} = 3$

Substituting the Value

Now that we have evaluated the expression inside the parentheses, we can substitute the value into the fraction.

$\frac{1}{(9^{\frac{1}{2}})^3} = \frac{1}{3^3}$

Evaluating the Expression Further

To evaluate the expression further, we need to simplify the fraction.

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

Conclusion

In conclusion, the expression $\left(9^{\frac{5}{2}}\right)^0 \left(9^{-\frac{3}{2}}\right)$ can be evaluated using the properties of exponents. By applying the zero exponent rule and the negative exponent rule, we can simplify the expression and evaluate it further. The final answer is $\frac{1}{27}$ .

Frequently Asked Questions

What is the zero exponent rule? The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1.
What is the negative exponent rule? The negative exponent rule states that $a^{-b} = \frac{1}{a^b}$ .
How do you simplify a fraction with an exponent? To simplify a fraction with an exponent, you need to evaluate the expression inside the parentheses and substitute the value into the fraction.

Final Answer

The final answer is $\boxed{\frac{1}{27}}$ .

Introduction

In our previous article, we evaluated the expression $\left(9^{\frac{5}{2}}\right)^0 \left(9^{-\frac{3}{2}}\right)$ using the properties of exponents. In this article, we will answer some frequently asked questions related to evaluating expressions with exponents.

Q&A

Q: What is the zero exponent rule?

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

Q: What is the negative exponent rule?

A: The negative exponent rule states that $a^{-b} = \frac{1}{a^b}$ . For example, $a^{-2} = \frac{1}{a^2}$ .

Q: How do you simplify a fraction with an exponent?

A: To simplify a fraction with an exponent, you need to evaluate the expression inside the parentheses and substitute the value into the fraction. For example, $\frac{1}{(9^{\frac{1}{2}})^3} = \frac{1}{3^3}$ .

Q: What is the order of operations when evaluating expressions with exponents?

A: The order of operations when evaluating expressions with exponents is:

Evaluate the expressions inside the parentheses.
Evaluate any exponents.
Multiply and divide from left to right.
Add and subtract from left to right.

Q: How do you handle negative exponents in expressions?

A: To handle negative exponents in expressions, you can use the negative exponent rule, which states that $a^{-b} = \frac{1}{a^b}$ . For example, $a^{-2} = \frac{1}{a^2}$ .

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents repeated multiplication, while a negative exponent represents repeated division. For example, $a^2 = a \times a$ and $a^{-2} = \frac{1}{a \times a}$ .

Q: How do you evaluate expressions with multiple exponents?

A: To evaluate expressions with multiple exponents, you need to follow the order of operations and apply the exponent rules. For example, $(a^2)^3 = a^{2 \times 3} = a^6$ .

Conclusion

In conclusion, evaluating expressions with exponents requires a good understanding of the exponent rules and the order of operations. By following the rules and applying the exponent properties, you can simplify complex expressions and arrive at the correct solution.

Frequently Asked Questions

What is the exponent rule for multiplying variables with exponents? The exponent rule for multiplying variables with exponents states that when multiplying variables with the same base, you add the exponents. For example, $a^2 \times a^3 = a^{2+3} = a^5$ .
What is the exponent rule for dividing variables with exponents? The exponent rule for dividing variables with exponents states that when dividing variables with the same base, you subtract the exponents. For example, $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$ .
How do you handle fractional exponents in expressions? To handle fractional exponents in expressions, you can use the property that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . For example, $a^{\frac{1}{2}} = \sqrt{a}$ .

Final Answer

The final answer is $\boxed{\frac{1}{27}}$ .