Simplify. Express your answer using positive exponents: (s^0 * t^(-1) * u^(-1)) / (s^(-7) * t * u^(-8))

Understand the Problem

The question is asking to simplify an expression involving exponents. The expression includes variables raised to different powers, and the user is required to express the answer using positive exponents.

Answer

The simplified expression is $s^7 t^2 u$.

Steps to Solve

Write the expression clearly The given expression is:

$$ \frac{s^0 t^{-1} u^{-1}}{s^{-7} t^u} $$

Simplify the numerator Recall that any variable raised to the power of 0 equals 1. Therefore, we have:

$$ s^0 = 1 $$

So the numerator simplifies to:

$$ t^{-1} u^{-1} $$

Rewrite the expression Now substitute back into the expression:

$$ \frac{t^{-1} u^{-1}}{s^{-7} t^1} $$

Combine the terms Rewrite the expression:

$$ \frac{t^{-1} u^{-1}}{s^{-7} t} = \frac{t^{-1}}{t} \cdot \frac{u^{-1}}{s^{-7}} $$

Apply the laws of exponents Using the rule that $a^{-m} = \frac{1}{a^m}$, we can simplify:

$$ t^{-1} \cdot t^{-1} = t^{-2} $$

This implies:

$$ = \frac{u^{-1}}{s^{-7}} \cdot t^{-2} $$

Rewrite using positive exponents By applying the property that $a^{-m} = \frac{1}{a^m}$, rewrite the expression:

$$ = \frac{1}{su^1} \cdot \frac{1}{t^2} $$

So, the final expression becomes:

$$ s^7 t^2 u $$

The simplified expression is:

$$ s^{7} t^{2} u $$

More Information

Using the division rule for exponents and understanding that any variable raised to zero equals one helps simplify the expression effectively.

Tips

Forgetting that $s^0 = 1$ can lead to an unnecessary complexity in the problem.
Mixing up the negative exponents and their positive counterparts.

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