Rewrite Using A Single Positive Exponent: ( 5 3 ) − 6 \left(5^3\right)^{-6} ( 5 3 ) − 6

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Understanding Exponents and Their Properties

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The exponentiation operation is denoted by a caret (^) or an exponentiation operator (e.g., **, ^). When dealing with exponents, it's essential to understand the properties and rules governing their behavior. In this article, we will focus on rewriting the expression $\left(5^3\right)^{-6}$ using a single positive exponent.

The Power of a Power Property

The power of a power property states that for any numbers $a$, $b$, and $c$, the following equation holds:

$$\left(a^b\right)^c = a^{b \cdot c}$$

This property allows us to simplify expressions involving exponents by multiplying the exponents. In the given expression $\left(5^3\right)^{-6}$, we can apply the power of a power property to rewrite it using a single positive exponent.

Rewriting the Expression

Using the power of a power property, we can rewrite the expression $\left(5^3\right)^{-6}$ as follows:

$$\left(5^3\right)^{-6} = 5^{3 \cdot (-6)}$$

Now, we can simplify the exponent by multiplying $3$ and $-6$:

$$5^{3 \cdot (-6)} = 5^{-18}$$

Understanding Negative Exponents

Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base. In the expression $5^{-18}$, the negative exponent indicates that we need to take the reciprocal of $5$ and raise it to the power of $18$.

Rewriting Negative Exponents as Positive Exponents

To rewrite a negative exponent as a positive exponent, we can use the following rule:

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

Applying this rule to the expression $5^{-18}$, we get:

$$5^{-18} = \frac{1}{5^{18}}$$

Simplifying the Expression

Now that we have rewritten the expression using a single positive exponent, we can simplify it further by evaluating the exponent. In this case, we need to calculate $5^{18}$.

Calculating $5^{18}$

To calculate $5^{18}$, we can use the fact that $5^2 = 25$ and $5^3 = 125$. We can then use these values to calculate $5^{18}$:

$$5^{18} = (5^2)^9 = 25^9$$

Now, we can calculate $25^9$:

$$25^9 = 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25$$

Using a calculator or a computer program, we can evaluate this expression:

$$25^9 = 244,140,625$$

Conclusion

In this article, we have rewritten the expression $\left(5^3\right)^{-6}$ using a single positive exponent. We have applied the power of a power property to simplify the expression and then rewritten the negative exponent as a positive exponent. Finally, we have simplified the expression by evaluating the exponent. The result is a simplified expression that can be used in various mathematical contexts.

Key Takeaways

• The power of a power property allows us to simplify expressions involving exponents by multiplying the exponents.
• Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base.
• The expression $\left(5^3\right)^{-6}$ can be rewritten as $5^{-18}$ using the power of a power property.
• The expression $5^{-18}$ can be rewritten as $\frac{1}{5^{18}}$ using the rule for negative exponents.
• The expression $\frac{1}{5^{18}}$ can be simplified by evaluating the exponent.

Further Reading

For more information on exponents and their properties, we recommend the following resources:

• Khan Academy: Exponents and Exponent Rules
• Math Is Fun: Exponents
• Wolfram MathWorld: Exponentiation

Understanding Exponents and Their Properties

In our previous article, we explored the concept of exponents and their properties. We learned how to rewrite expressions using a single positive exponent and how to simplify expressions involving exponents. In this article, we will answer some frequently asked questions about exponents and their properties.

Q: What is an exponent?

A: An exponent is a number that represents repeated multiplication of a base number. For example, $5^3$ means $5$ multiplied by itself $3$ times, or $5 \cdot 5 \cdot 5$.

Q: What is the power of a power property?

A: The power of a power property states that for any numbers $a$, $b$, and $c$, the following equation holds:

$$\left(a^b\right)^c = a^{b \cdot c}$$

This property allows us to simplify expressions involving exponents by multiplying the exponents.

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

A: To rewrite a negative exponent as a positive exponent, you can use the following rule:

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

For example, $5^{-3}$ can be rewritten as $\frac{1}{5^3}$.

Q: What is the difference between $5^3$ and $5^{3^2}$?

A: $5^3$ means $5$ multiplied by itself $3$ times, or $5 \cdot 5 \cdot 5$. On the other hand, $5^{3^2}$ means $5$ multiplied by itself $3^2$ times, or $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$.

Q: How do I simplify an expression involving exponents?

A: To simplify an expression involving exponents, you can use the power of a power property and the rule for negative exponents. For example, $\left(5^3\right)^{-6}$ can be simplified as follows:

$$\left(5^3\right)^{-6} = 5^{3 \cdot (-6)} = 5^{-18} = \frac{1}{5^{18}}$$

Q: What is the value of $5^{18}$?

A: $5^{18}$ is equal to $244,140,625$.

Q: Can I use exponents with fractions?

A: Yes, you can use exponents with fractions. For example, $\left(\frac{1}{2}\right)^3$ means $\frac{1}{2}$ multiplied by itself $3$ times, or $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}$.

Q: How do I evaluate an expression involving exponents with fractions?

A: To evaluate an expression involving exponents with fractions, you can use the power of a power property and the rule for negative exponents. For example, $\left(\frac{1}{2}\right)^{-3}$ can be evaluated as follows:

$$\left(\frac{1}{2}\right)^{-3} = \frac{1}{\left(\frac{1}{2}\right)^3} = \frac{1}{\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}} = \frac{1}{\frac{1}{8}} = 8$$

Conclusion

In this article, we have answered some frequently asked questions about exponents and their properties. We have learned how to rewrite expressions using a single positive exponent, how to simplify expressions involving exponents, and how to evaluate expressions involving exponents with fractions. By following the steps outlined in this article, you can master the concept of exponents and their properties.

Key Takeaways

• Exponents are a fundamental concept in mathematics that represent repeated multiplication of a base number.
• The power of a power property allows us to simplify expressions involving exponents by multiplying the exponents.
• Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base.
• Exponents can be used with fractions, and the power of a power property and the rule for negative exponents can be used to evaluate expressions involving exponents with fractions.

Further Reading

For more information on exponents and their properties, we recommend the following resources:

• Khan Academy: Exponents and Exponent Rules
• Math Is Fun: Exponents
• Wolfram MathWorld: Exponentiation

By mastering the concept of exponents and their properties, you can solve a wide range of mathematical problems and become a proficient mathematician.