Simplify The Expression. Express Your Answer Using Positive Exponents.$\[ \frac{8 T^{-14} U^0 V^{-3} \cdot 10 T^{-1} U^{-1} V^{-1}}{t^{58} U^{-1} V^0} \\]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying mathematical concepts. In this article, we will focus on simplifying a given expression using positive exponents. We will break down the expression into smaller parts, apply the rules of exponents, and finally arrive at the simplified form.

The Given Expression

The given expression is:

8t−14u0v−3⋅10t−1u−1v−1t58u−1v0\frac{8 t^{-14} u^0 v^{-3} \cdot 10 t^{-1} u^{-1} v^{-1}}{t^{58} u^{-1} v^0}

Step 1: Apply the Product Rule for Exponents

The product rule for exponents states that when we multiply two numbers with the same base, we add their exponents. In this case, we have two numbers with the same base, tt, and another number with the same base, uu. We can apply the product rule to simplify the expression.

8t−14u0v−3⋅10t−1u−1v−1t58u−1v0=8⋅10t−14t−1u0u−1v−3v−1t58u−1v0\frac{8 t^{-14} u^0 v^{-3} \cdot 10 t^{-1} u^{-1} v^{-1}}{t^{58} u^{-1} v^0} = \frac{8 \cdot 10 t^{-14} t^{-1} u^0 u^{-1} v^{-3} v^{-1}}{t^{58} u^{-1} v^0}

Step 2: Simplify the Numerator

Using the product rule, we can simplify the numerator by adding the exponents of the same base.

8⋅10t−14t−1u0u−1v−3v−1=80t−15u−1v−48 \cdot 10 t^{-14} t^{-1} u^0 u^{-1} v^{-3} v^{-1} = 80 t^{-15} u^{-1} v^{-4}

Step 3: Simplify the Denominator

The denominator is already simplified, but we can rewrite it using positive exponents.

t58u−1v0=t58u1v0=t58uv0=t58ut^{58} u^{-1} v^0 = \frac{t^{58}}{u^1 v^0} = \frac{t^{58}}{u v^0} = \frac{t^{58}}{u}

Step 4: Simplify the Expression

Now that we have simplified the numerator and the denominator, we can rewrite the expression using positive exponents.

80t−15u−1v−4t58u=80t−15u−1v−4⋅ut58\frac{80 t^{-15} u^{-1} v^{-4}}{\frac{t^{58}}{u}} = 80 t^{-15} u^{-1} v^{-4} \cdot \frac{u}{t^{58}}

Step 5: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that when we divide two numbers with the same base, we subtract their exponents. In this case, we have two numbers with the same base, tt, and another number with the same base, uu. We can apply the quotient rule to simplify the expression.

80t−15u−1v−4⋅ut58=80t−15t−58u−1uv−480 t^{-15} u^{-1} v^{-4} \cdot \frac{u}{t^{58}} = 80 t^{-15} t^{-58} u^{-1} u v^{-4}

Step 6: Simplify the Expression

Using the quotient rule, we can simplify the expression by subtracting the exponents of the same base.

80t−15t−58u−1uv−4=80t−73u0v−480 t^{-15} t^{-58} u^{-1} u v^{-4} = 80 t^{-73} u^0 v^{-4}

Step 7: Simplify the Expression

Finally, we can simplify the expression by removing the zero exponent.

80t−73u0v−4=80t−73v−480 t^{-73} u^0 v^{-4} = 80 t^{-73} v^{-4}

Conclusion

In this article, we simplified the given expression using positive exponents. We applied the product rule, quotient rule, and removed the zero exponent to arrive at the final simplified form. This example demonstrates the importance of understanding the rules of exponents and how to apply them to simplify complex expressions.

Final Answer

The final answer is:

80t−73v−480 t^{-73} v^{-4}

Discussion

This problem requires a deep understanding of the rules of exponents and how to apply them to simplify complex expressions. The product rule and quotient rule are essential concepts in algebra that help us simplify expressions and solve problems. By following the steps outlined in this article, we can simplify any expression using positive exponents.

Additional Resources

For more information on simplifying expressions using positive exponents, please refer to the following resources:

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Q&A: Simplifying Expressions Using Positive Exponents

Q: What are the rules of exponents?

A: The rules of exponents are a set of mathematical rules that help us simplify expressions with exponents. The main rules are:

  • Product Rule: When we multiply two numbers with the same base, we add their exponents.
  • Quotient Rule: When we divide two numbers with the same base, we subtract their exponents.
  • Power Rule: When we raise a power to a power, we multiply the exponents.

Q: How do I simplify an expression using positive exponents?

A: To simplify an expression using positive exponents, follow these steps:

  1. Apply the Product Rule: Multiply the numbers with the same base and add their exponents.
  2. Apply the Quotient Rule: Divide the numbers with the same base and subtract their exponents.
  3. Remove Zero Exponents: Remove any zero exponents from the expression.
  4. Simplify the Expression: Simplify the expression by combining like terms.

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

A: A positive exponent is a number that is raised to a power, such as 232^3. A negative exponent is a number that is raised to a power, but with a negative sign, such as 2−32^{-3}. When we simplify an expression with negative exponents, we can rewrite it using positive exponents by flipping the fraction.

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

A: To rewrite a negative exponent as a positive exponent, follow these steps:

  1. Flip the Fraction: Flip the fraction by swapping the numerator and denominator.
  2. Change the Sign: Change the sign of the exponent to positive.

For example, 2−3=1232^{-3} = \frac{1}{2^3}.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is:

80t−73v−480 t^{-73} v^{-4}

However, we can rewrite this expression using positive exponents by flipping the fraction:

80t73v4\frac{80}{t^{73} v^4}

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Forgetting to apply the Product Rule: Make sure to multiply the numbers with the same base and add their exponents.
  • Forgetting to apply the Quotient Rule: Make sure to divide the numbers with the same base and subtract their exponents.
  • Not removing zero exponents: Make sure to remove any zero exponents from the expression.
  • Not simplifying the expression: Make sure to simplify the expression by combining like terms.

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

A: You can practice simplifying expressions using positive exponents by:

  • Working through practice problems: Try simplifying expressions with different bases and exponents.
  • Using online resources: Use online resources, such as Khan Academy or Mathway, to practice simplifying expressions.
  • Asking a teacher or tutor: Ask a teacher or tutor for help and guidance.

By following these steps and practicing regularly, you can become proficient in simplifying expressions using positive exponents.