Simplify The Expression: $\frac{9}{h^{-3}}$

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Understanding the Problem

The given expression is $\frac{9}{h^{-3}}$. This expression involves a negative exponent, which can be simplified using the properties of exponents. In this article, we will simplify the given expression and provide a step-by-step solution.

Properties of Exponents

Before we simplify the expression, let's review the properties of exponents. The property of negative exponents states that $a^{-n} = \frac{1}{a^n}$. This property allows us to rewrite negative exponents as fractions.

Simplifying the Expression

Using the property of negative exponents, we can rewrite the given expression as follows:

$$\frac{9}{h^{-3}} = \frac{1}{\frac{9}{h^3}}$$

Now, let's simplify the expression further. We can rewrite the fraction as a product of two fractions:

$$\frac{1}{\frac{9}{h^3}} = \frac{1}{9} \cdot \frac{h^3}{1}$$

Using the property of multiplication, we can rewrite the expression as:

$$\frac{1}{9} \cdot \frac{h^3}{1} = \frac{h^3}{9}$$

Therefore, the simplified expression is $\frac{h^3}{9}$.

Conclusion

In this article, we simplified the expression $\frac{9}{h^{-3}}$ using the properties of exponents. We rewrote the negative exponent as a fraction and then simplified the expression further using the property of multiplication. The final simplified expression is $\frac{h^3}{9}$.

Example Use Cases

The simplified expression $\frac{h^3}{9}$ can be used in various mathematical applications, such as:

• Calculating the volume of a sphere: The volume of a sphere is given by the formula $\frac{4}{3}\pi r^3$. If we want to calculate the volume of a sphere with radius $h$, we can use the simplified expression $\frac{h^3}{9}$.
• Solving equations involving exponents: The simplified expression $\frac{h^3}{9}$ can be used to solve equations involving exponents, such as $h^{-3} = 9$.

Tips and Tricks

When simplifying expressions involving negative exponents, remember to use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. This property allows us to rewrite negative exponents as fractions.

Common Mistakes

When simplifying expressions involving negative exponents, some common mistakes to avoid include:

• Not using the property of negative exponents
• Not rewriting negative exponents as fractions
• Not simplifying the expression further using the property of multiplication

Conclusion

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Q: What is a negative exponent?

A: A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. For example, $a^{-n}$ is a negative exponent, where $a$ is a number and $n$ is a positive integer.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. This property allows you to rewrite negative exponents as fractions.

Q: What is the property of negative exponents?

A: The property of negative exponents states that $a^{-n} = \frac{1}{a^n}$. This property allows you to rewrite negative exponents as fractions.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, you can use the property of negative exponents. For example, if you have the expression $a^{-n}$, you can rewrite it as $\frac{1}{a^n}$.

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

A: Yes, you can simplify an expression with a negative exponent further using the property of multiplication. For example, if you have the expression $\frac{1}{a^n}$, you can rewrite it as $\frac{a^{-n}}{1}$.

Q: What is the final simplified expression for $\frac{9}{h^{-3}}$?

A: The final simplified expression for $\frac{9}{h^{-3}}$ is $\frac{h^3}{9}$.

Q: Can I use the simplified expression $\frac{h^3}{9}$ in real-world applications?

A: Yes, you can use the simplified expression $\frac{h^3}{9}$ in real-world applications, such as calculating the volume of a sphere or solving equations involving exponents.

Q: What are some common mistakes to avoid when simplifying expressions with negative exponents?

A: Some common mistakes to avoid when simplifying expressions with negative exponents include:

• Not using the property of negative exponents
• Not rewriting negative exponents as fractions
• Not simplifying the expression further using the property of multiplication

Q: How can I practice simplifying expressions with negative exponents?

A: You can practice simplifying expressions with negative exponents by working through examples and exercises in a math textbook or online resource. You can also try simplifying expressions with negative exponents on your own using the property of negative exponents.

Q: What are some real-world applications of simplifying expressions with negative exponents?

A: Some real-world applications of simplifying expressions with negative exponents include:

• Calculating the volume of a sphere
• Solving equations involving exponents
• Working with scientific notation

Q: Can I use the property of negative exponents to simplify expressions with multiple negative exponents?

A: Yes, you can use the property of negative exponents to simplify expressions with multiple negative exponents. For example, if you have the expression $a^{-n}b^{-m}$, you can rewrite it as $\frac{1}{a^n}\frac{1}{b^m}$.

Q: How can I simplify an expression with multiple negative exponents?

A: To simplify an expression with multiple negative exponents, you can use the property of negative exponents to rewrite each negative exponent as a fraction, and then simplify the expression further using the property of multiplication.