Problem 61 Simplify. Write the answers with... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 61

Simplify. Write the answers with positive exponents only. $$\frac{t^{4}}{t^{9}}$$

Short Answer

Expert verified

$ \frac{t^4}{t^9} = \frac{1}{t^5} $

Step by step solution

- Apply the Quotient Rule

Use the quotient rule which states that when you divide powers with the same base, you subtract the exponents: \[ \frac{t^a}{t^b} = t^{a-b} \] In this case, $a = 4$ and $b = 9$, so: \[ \frac{t^4}{t^9} = t^{4-9} = t^{-5} \]

- Convert Negative Exponents to Positive

Recall that a negative exponent signifies a reciprocal: \[ t^{-n} = \frac{1}{t^n} \] For $ t^{-5} $, this conversion gives: \[ t^{-5} = \frac{1}{t^5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule

When simplifying expressions with exponents, one of the most useful rules is the quotient rule.
This rule is applied when you are dividing powers that have the same base.
The formula for the quotient rule is: \[ \frac{t^a}{t^b} = t^{a-b} \]
Here is a simple example:

$\frac{t^5}{t^2} = t^{5-2} = t^3 $

It is crucial to remember that this rule only works if the bases are the same.
In our original exercise, we had $\frac{t^4}{t^9}$. By applying the quotient rule, we get $ t^{4-9} = t^{-5}$.

Positive Exponents

In mathematics, we often prefer to write expressions using positive exponents.
Positive exponents are easier to interpret and manipulate.
A positive exponent tells you how many times to multiply a base.
For instance, $ t^3 $ means $ t \times t \times t $.

If you see $ t^{4} $, it means $ t \times t \times t \times t $ 鈥� the base $\text{t}$ multiplied by itself 4 times.

In our earlier steps, we arrived at $ t^{-5} $.
This is a valid result, but we prefer to express it with a positive exponent.

Negative Exponents

Negative exponents can initially seem confusing, but they simply indicate reciprocals.
The rule for negative exponents is: \[ t^{-n} = \frac{1}{t^n} \]
Here, $ t^{-5} $ is converted to $ \frac{1}{t^5} $.
This means that $ t^{-5} $ is the same as $ \frac{1}{t \times t \times t \times t \times t} $.

Another example would be $ t^{-2} = \frac{1}{t^2} = \frac{1}{t \times t} $.

Recognizing and converting negative exponents to positive exponents is key in simplifying your expressions.
In the end, our original expression $ \frac{t^4}{t^9} $ simplifies to $ \frac{1}{t^5} $.

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91影视

Simplify. Write the answers with positive exponents only. $$\frac{t^{4}}{t^{9}}$$

Short Answer

Step by step solution

- Apply the Quotient Rule

- Convert Negative Exponents to Positive

Key Concepts

Quotient Rule

Positive Exponents

Negative Exponents

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