Simplify. Write The Expression Using Only Positive Exponents.$10t^{-5}$The Simplified Expression Is $\square$.

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Understanding the Problem

When dealing with exponents, it's essential to simplify expressions to make them easier to work with. In this case, we're given the expression 10t−510t^{-5} and asked to rewrite it using only positive exponents. This means we need to eliminate the negative exponent and express the expression in a more manageable form.

What are Exponents?

Before we dive into simplifying the expression, let's quickly review what exponents are. An exponent is a small number that is raised to the power of a larger number. In the expression t−5t^{-5}, the exponent -5 indicates that the base tt is being raised to the power of -5. This means that we're essentially dividing 1 by tt five times.

Simplifying Negative Exponents

To simplify an expression with a negative exponent, we can use the rule that states a−n=1ana^{-n} = \frac{1}{a^n}. This means that we can rewrite the expression t−5t^{-5} as 1t5\frac{1}{t^5}.

Applying the Rule to the Original Expression

Now that we've simplified the expression with the negative exponent, we can apply this rule to the original expression 10t−510t^{-5}. By substituting 1t5\frac{1}{t^5} for t−5t^{-5}, we get:

10t−5=10⋅1t510t^{-5} = 10 \cdot \frac{1}{t^5}

Simplifying the Expression Further

To simplify the expression further, we can multiply the numerator and denominator by t5t^5 to get rid of the fraction. This gives us:

10⋅1t5=10t510 \cdot \frac{1}{t^5} = \frac{10}{t^5}

The Final Simplified Expression

The final simplified expression is 10t5\boxed{\frac{10}{t^5}}. This expression uses only positive exponents and is easier to work with than the original expression.

Conclusion

Simplifying expressions with negative exponents is an essential skill in mathematics. By applying the rule that a−n=1ana^{-n} = \frac{1}{a^n}, we can rewrite expressions with negative exponents in a more manageable form. In this case, we simplified the expression 10t−510t^{-5} to 10t5\frac{10}{t^5}, which uses only positive exponents.

Tips and Tricks

  • When dealing with negative exponents, remember that a−n=1ana^{-n} = \frac{1}{a^n}.
  • To simplify an expression with a negative exponent, substitute 1an\frac{1}{a^n} for a−na^{-n}.
  • When multiplying or dividing expressions with exponents, remember to multiply or divide the exponents as well.

Common Mistakes to Avoid

  • Don't forget to apply the rule that a−n=1ana^{-n} = \frac{1}{a^n} when dealing with negative exponents.
  • Be careful when multiplying or dividing expressions with exponents, as this can lead to errors.
  • Make sure to simplify expressions fully to avoid confusion.

Real-World Applications

Simplifying expressions with negative exponents has many real-world applications. For example, in physics, we often deal with expressions that involve negative exponents, such as the equation for the force of gravity. By simplifying these expressions, we can better understand the underlying physics and make more accurate predictions.

Practice Problems

  • Simplify the expression 2x−32x^{-3}.
  • Simplify the expression 1y4\frac{1}{y^4}.
  • Simplify the expression 3z−23z^{-2}.

Solutions to Practice Problems

  • 2x−3=2x32x^{-3} = \frac{2}{x^3}
  • 1y4=y−4\frac{1}{y^4} = y^{-4}
  • 3z−2=3z23z^{-2} = \frac{3}{z^2}

Frequently Asked Questions

We've covered the basics of simplifying expressions with negative exponents, but we know that you may still have some questions. Here are some frequently asked questions and their answers:

Q: What is the rule for simplifying negative exponents?

A: The rule for simplifying negative exponents is that a−n=1ana^{-n} = \frac{1}{a^n}. This means that we can rewrite an expression with a negative exponent as a fraction with the base raised to the power of the positive exponent.

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

A: To simplify an expression with a negative exponent, substitute 1an\frac{1}{a^n} for a−na^{-n}. For example, if we have the expression 2x−32x^{-3}, we can simplify it by substituting 1x3\frac{1}{x^3} for x−3x^{-3}.

Q: What if I have a negative exponent in the denominator?

A: If you have a negative exponent in the denominator, you can simplify it by moving the base to the numerator and changing the sign of the exponent. For example, if we have the expression 1y−4\frac{1}{y^{-4}}, we can simplify it by moving the base to the numerator and changing the sign of the exponent to get y4y^4.

Q: Can I simplify an expression with a negative exponent by multiplying or dividing?

A: Yes, you can simplify an expression with a negative exponent by multiplying or dividing. For example, if we have the expression 2x−32x^{-3}, we can simplify it by multiplying both the numerator and denominator by x3x^3 to get 2x3x6\frac{2x^3}{x^6}.

Q: What if I have a negative exponent in the numerator and denominator?

A: If you have a negative exponent in both the numerator and denominator, you can simplify it by canceling out the bases and changing the sign of the exponents. For example, if we have the expression y−4y−3\frac{y^{-4}}{y^{-3}}, we can simplify it by canceling out the bases and changing the sign of the exponents to get yy.

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

A: Yes, you can simplify an expression with a negative exponent using a calculator. However, keep in mind that calculators may not always display the simplified form of the expression.

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:

  • Forgetting to apply the rule that a−n=1ana^{-n} = \frac{1}{a^n}.
  • Not simplifying the expression fully.
  • Making errors when multiplying or dividing expressions with exponents.

Real-World Applications of Simplifying Expressions with Negative Exponents

Simplifying expressions with negative exponents has many real-world applications. Here are a few examples:

  • Physics: In physics, we often deal with expressions that involve negative exponents, such as the equation for the force of gravity. By simplifying these expressions, we can better understand the underlying physics and make more accurate predictions.
  • Engineering: In engineering, we often deal with expressions that involve negative exponents, such as the equation for the stress on a material. By simplifying these expressions, we can better understand the underlying physics and make more accurate predictions.
  • Computer Science: In computer science, we often deal with expressions that involve negative exponents, such as the equation for the time complexity of an algorithm. By simplifying these expressions, we can better understand the underlying physics and make more accurate predictions.

Practice Problems

  • Simplify the expression 3x−23x^{-2}.
  • Simplify the expression 1y−3\frac{1}{y^{-3}}.
  • Simplify the expression 2z−42z^{-4}.

Solutions to Practice Problems

  • 3x−2=3x23x^{-2} = \frac{3}{x^2}
  • 1y−3=y3\frac{1}{y^{-3}} = y^3
  • 2z−4=2z42z^{-4} = \frac{2}{z^4}