Simplify. Express Your Answer Using Positive Exponents.${\frac{\left(2 Z^0\right)(2 Z)}{z^0}}$

Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In the expression $\frac{\left(2 z^0\right)(2 z)}{z^0}$, we are dealing with exponents and their properties. To simplify this expression, we need to understand the rules of exponents and apply them correctly.

The Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, this can be represented as $a^0 = 1$, where $a$ is any non-zero number. In the given expression, we have $z^0$ in the numerator and denominator. According to the zero exponent rule, both $z^0$ terms are equal to 1.

Simplifying the Expression

Now that we have applied the zero exponent rule, the expression becomes $\frac{\left(2 \cdot 1\right)(2 z)}{1}$. We can simplify this further by multiplying the numbers in the numerator. This gives us $\frac{4 z}{1}$.

Applying the Rule of Negative Exponents

In the expression $\frac{4 z}{1}$, we can rewrite the denominator as $1$ raised to the power of 0, which is equal to 1. However, we can also rewrite the expression using a negative exponent. The rule of negative exponents states that $a^{-n} = \frac{1}{a^n}$. In this case, we can rewrite the expression as $4 z \cdot z^{-1}$.

Simplifying the Expression Using Positive Exponents

Now that we have applied the rule of negative exponents, we can simplify the expression further. We can rewrite $z^{-1}$ as $\frac{1}{z}$, which gives us $4 z \cdot \frac{1}{z}$. We can simplify this further by canceling out the $z$ terms, which gives us $4$.

Conclusion

In conclusion, we have simplified the expression $\frac{\left(2 z^0\right)(2 z)}{z^0}$ using positive exponents. We applied the zero exponent rule to simplify the expression, and then applied the rule of negative exponents to rewrite the expression in a different form. Finally, we simplified the expression further using positive exponents.

Key Takeaways

• The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1.
• The rule of negative exponents states that $a^{-n} = \frac{1}{a^n}$.
• We can simplify expressions using positive exponents by applying the rules of exponents.

Examples and Applications

• Simplify the expression $\frac{\left(3 x^0\right)(3 x)}{x^0}$ using positive exponents.
• Simplify the expression $\frac{\left(2 y^0\right)(2 y)}{y^0}$ using positive exponents.
• Apply the rule of negative exponents to simplify the expression $a^{-n}$.

Further Reading

• Exponents and their properties
• Rules of exponents
• Simplifying expressions using positive exponents

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Rules of Exponents" by Mathway
• [3] "Simplifying Expressions Using Positive Exponents" by Khan Academy
  

Frequently Asked Questions

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. Mathematically, this can be represented as $a^0 = 1$, where $a$ is any non-zero number.

Q: How do I apply the zero exponent rule to simplify an expression?

A: To apply the zero exponent rule, simply replace any non-zero number raised to the power of zero with 1. For example, if you have the expression $\frac{\left(2 z^0\right)(2 z)}{z^0}$, you can simplify it by replacing the $z^0$ terms with 1.

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that any number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power.

Q: How do I apply the rule of negative exponents to simplify an expression?

A: To apply the rule of negative exponents, simply rewrite any number raised to a negative power as the reciprocal of the number raised to the positive power. For example, if you have the expression $a^{-n}$, you can rewrite it as $\frac{1}{a^n}$.

Q: Can I simplify expressions using positive exponents?

A: Yes, you can simplify expressions using positive exponents. By applying the rules of exponents, you can rewrite expressions in a simpler form using positive exponents.

Q: What are some examples of expressions that can be simplified using positive exponents?

A: Some examples of expressions that can be simplified using positive exponents include:

• $\frac{\left(2 z^0\right)(2 z)}{z^0}$
• $\frac{\left(3 x^0\right)(3 x)}{x^0}$
• $\frac{\left(2 y^0\right)(2 y)}{y^0}$

Q: How do I know when to use positive exponents to simplify an expression?

A: You should use positive exponents to simplify an expression when the expression contains numbers or variables raised to a power. By applying the rules of exponents, you can rewrite the expression in a simpler form using positive exponents.

Q: What are some common mistakes to avoid when simplifying expressions using positive exponents?

A: Some common mistakes to avoid when simplifying expressions using positive exponents include:

• Forgetting to apply the zero exponent rule
• Forgetting to apply the rule of negative exponents
• Not simplifying the expression enough

Q: How can I practice simplifying expressions using positive exponents?

A: You can practice simplifying expressions using positive exponents by working through examples and exercises. You can also try simplifying expressions on your own and then checking your work to make sure you got it right.

Additional Resources

• Exponents and Exponential Functions by Math Open Reference
• Rules of Exponents by Mathway
• Simplifying Expressions Using Positive Exponents by Khan Academy

Conclusion

Simplifying expressions using positive exponents is an important skill to have in mathematics. By applying the rules of exponents, you can rewrite expressions in a simpler form using positive exponents. Remember to apply the zero exponent rule and the rule of negative exponents to simplify expressions, and practice simplifying expressions using positive exponents to become more confident in your skills.