Rewrite Each Expression Using Only Positive Exponents.a. $39^{-5}\left(39^{-6} \cdot 39\right)^{-2}$b. $\frac{11^{-3} \cdot 40^{-2}}{40^6 \cdot 11^{-8}}$c. $16^{-10} \cdot 16^4\left(16^{-5}\right)^2$d.

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In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. However, when dealing with negative exponents, it can be challenging to simplify expressions. In this article, we will explore how to rewrite each expression using only positive exponents.

Rewriting Negative Exponents

Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base and changing the sign of the exponent. This is based on the rule that a−n=1ana^{-n} = \frac{1}{a^n}.

a. 39−5(39−6⋅39)−239^{-5}\left(39^{-6} \cdot 39\right)^{-2}

To rewrite this expression using only positive exponents, we can start by simplifying the expression inside the parentheses.

(39−6⋅39)−2=(39−6+1)−2=(39−5)−2\left(39^{-6} \cdot 39\right)^{-2} = \left(39^{-6+1}\right)^{-2} = \left(39^{-5}\right)^{-2}

Now, we can rewrite the expression using the rule for negative exponents.

39−5(39−6⋅39)−2=39−5⋅(1395)−239^{-5}\left(39^{-6} \cdot 39\right)^{-2} = 39^{-5} \cdot \left(\frac{1}{39^5}\right)^{-2}

Using the rule for negative exponents again, we can rewrite the expression as:

39−5⋅(1395)−2=39−5⋅391039^{-5} \cdot \left(\frac{1}{39^5}\right)^{-2} = 39^{-5} \cdot 39^{10}

Finally, we can simplify the expression by combining the exponents.

39−5⋅3910=3910−5=39539^{-5} \cdot 39^{10} = 39^{10-5} = 39^5

b. 11−3⋅40−2406⋅11−8\frac{11^{-3} \cdot 40^{-2}}{40^6 \cdot 11^{-8}}

To rewrite this expression using only positive exponents, we can start by simplifying the numerator and denominator separately.

11−3⋅40−2406⋅11−8=1113⋅402⋅40−6118\frac{11^{-3} \cdot 40^{-2}}{40^6 \cdot 11^{-8}} = \frac{1}{11^3 \cdot 40^2} \cdot \frac{40^{-6}}{11^8}

Now, we can rewrite the expression using the rule for negative exponents.

1113⋅402⋅40−6118=1113⋅402⋅1118⋅406\frac{1}{11^3 \cdot 40^2} \cdot \frac{40^{-6}}{11^8} = \frac{1}{11^3 \cdot 40^2} \cdot \frac{1}{11^8} \cdot 40^6

Using the rule for negative exponents again, we can rewrite the expression as:

1113⋅402⋅1118⋅406=1113+8⋅402−6\frac{1}{11^3 \cdot 40^2} \cdot \frac{1}{11^8} \cdot 40^6 = \frac{1}{11^{3+8} \cdot 40^{2-6}}

Finally, we can simplify the expression by combining the exponents.

1113+8⋅402−6=11111⋅40−4=4041111\frac{1}{11^{3+8} \cdot 40^{2-6}} = \frac{1}{11^{11} \cdot 40^{-4}} = \frac{40^4}{11^{11}}

c. 16−10⋅164(16−5)216^{-10} \cdot 16^4\left(16^{-5}\right)^2

To rewrite this expression using only positive exponents, we can start by simplifying the expression inside the parentheses.

(16−5)2=16−5⋅2=16−10\left(16^{-5}\right)^2 = 16^{-5 \cdot 2} = 16^{-10}

Now, we can rewrite the expression using the rule for negative exponents.

16−10⋅164(16−5)2=16−10⋅164⋅16−1016^{-10} \cdot 16^4\left(16^{-5}\right)^2 = 16^{-10} \cdot 16^4 \cdot 16^{-10}

Using the rule for negative exponents again, we can rewrite the expression as:

16−10⋅164⋅16−10=11610⋅164⋅1161016^{-10} \cdot 16^4 \cdot 16^{-10} = \frac{1}{16^{10}} \cdot 16^4 \cdot \frac{1}{16^{10}}

Finally, we can simplify the expression by combining the exponents.

11610⋅164⋅11610=1641620=164−20=16−16\frac{1}{16^{10}} \cdot 16^4 \cdot \frac{1}{16^{10}} = \frac{16^4}{16^{20}} = 16^{4-20} = 16^{-16}

d. 16−10⋅16416^{-10} \cdot 16^4

To rewrite this expression using only positive exponents, we can use the rule for negative exponents.

16−10⋅164=11610⋅16416^{-10} \cdot 16^4 = \frac{1}{16^{10}} \cdot 16^4

Using the rule for negative exponents again, we can rewrite the expression as:

11610⋅164=1641610=164−10=16−6\frac{1}{16^{10}} \cdot 16^4 = \frac{16^4}{16^{10}} = 16^{4-10} = 16^{-6}

However, we can simplify the expression further by combining the exponents.

16−6=116616^{-6} = \frac{1}{16^6}

In our previous article, we explored how to rewrite expressions using only positive exponents. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A section to help you better understand how to rewrite expressions using only positive exponents.

Q: What is the rule for rewriting negative exponents?

A: The rule for rewriting negative exponents is to take the reciprocal of the base and change the sign of the exponent. This can be represented as a−n=1ana^{-n} = \frac{1}{a^n}.

Q: How do I rewrite an expression with a negative exponent in the numerator and a positive exponent in the denominator?

A: To rewrite an expression with a negative exponent in the numerator and a positive exponent in the denominator, you can start by simplifying the expression. Then, use the rule for negative exponents to rewrite the expression with only positive exponents.

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

A: A negative exponent represents a fraction, while a positive exponent represents a whole number. For example, a−na^{-n} represents 1an\frac{1}{a^n}, while ana^n represents aa multiplied by itself nn times.

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

A: To simplify an expression with multiple negative exponents, you can start by rewriting each negative exponent as a positive exponent using the rule for negative exponents. Then, combine the exponents by adding or subtracting them.

Q: Can I rewrite an expression with a negative exponent in the denominator?

A: Yes, you can rewrite an expression with a negative exponent in the denominator by taking the reciprocal of the base and changing the sign of the exponent. This can be represented as 1a−n=an\frac{1}{a^{-n}} = a^n.

Q: How do I rewrite an expression with a negative exponent in the numerator and a negative exponent in the denominator?

A: To rewrite an expression with a negative exponent in the numerator and a negative exponent in the denominator, you can start by simplifying the expression. Then, use the rule for negative exponents to rewrite the expression with only positive exponents.

Q: What is the final answer to the expression 16−10⋅164(16−5)216^{-10} \cdot 16^4\left(16^{-5}\right)^2?

A: The final answer to the expression 16−10⋅164(16−5)216^{-10} \cdot 16^4\left(16^{-5}\right)^2 is 16−1616^{-16}.

Q: What is the final answer to the expression 11−3⋅40−2406⋅11−8\frac{11^{-3} \cdot 40^{-2}}{40^6 \cdot 11^{-8}}?

A: The final answer to the expression 11−3⋅40−2406⋅11−8\frac{11^{-3} \cdot 40^{-2}}{40^6 \cdot 11^{-8}} is 4041111\frac{40^4}{11^{11}}.

Q: What is the final answer to the expression 16−10⋅16416^{-10} \cdot 16^4?

A: The final answer to the expression 16−10⋅16416^{-10} \cdot 16^4 is 1166\frac{1}{16^6}.

We hope this Q&A section has helped you better understand how to rewrite expressions using only positive exponents. Remember to practice, practice, practice, and you will become a pro at rewriting expressions in no time!