Problem 389 Simplify. $$\left(\frac{m^{4}}... [FREE SOLUTION]

Chapter 6: Problem 389

Simplify. $$\left(\frac{m^{4}}{m^{7}}\right)^{4}$$

Short Answer

Expert verified

The simplified expression is $m^{-12}$ or $\frac{1}{m^{12}}$.

Step by step solution

Simplify the Inner Fraction

Look at the fraction $\frac{m^{4}}{m^{7}}$. When dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator: $\frac{m^{4}}{m^{7}} = m^{4-7} = m^{-3}$.

Apply the Outer Exponent

Now take the simplified base from Step 1, $m^{-3}$, and apply the outer exponent $(m^{-3})^4$. When raising a power to a power, multiply the exponents: $m^{-3 \times 4} = m^{-12}$.

Simplify the Final Expression

The expression $m^{-12}$ is already simplified. In cases where you need positive exponents, it can be rewritten as $\frac{1}{m^{12}}$.

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Key Concepts

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

Division of Exponents

Whenever you come across a problem that involves dividing exponents with the same base, remember this simple rule: subtract the exponent in the denominator from the exponent in the numerator. Let's break it down using our example:

\[\frac{m^{4}}{m^{7}}\]

Here, both terms have the same base, which is 'm'. So, we subtract the exponent of the denominator from the exponent of the numerator:

\[\frac{m^{4}}{m^{7}} = m^{4-7} = m^{-3}\]

This process makes our work tons easier and helps us simplify complex expressions without breaking a sweat.

Negative Exponents

Dealing with negative exponents might seem tricky at first, but it's actually quite simple. Just remember this: a negative exponent means you take the reciprocal of the base and make the exponent positive. Here's how we apply this rule:

In the example we already simplified to \[m^{-3}\], the negative exponent tells us to write it like this:

\[m^{-3} = \frac{1}{m^{3}}\]

So, whenever you encounter a negative exponent, transform it into a positive exponent by moving the base to the denominator, or if it's already in a fraction, to the numerator.

Raising Powers to Powers

When raising a power to another power, all you need to do is multiply the exponents. Suppose we have something like this:

\[(m^{-3})^{4}\]

To simplify, you simply multiply the inner exponent by the outer exponent:

\[m^{-3 \times 4} = m^{-12}\]

It's straightforward鈥攋ust multiply. And don鈥檛 forget that a negative exponent changes the location in the fraction if you desire a positive exponent:

\[m^{-12} = \frac{1}{m^{12}}\]

These simple steps will help you tackle similar problems with ease and confidence.

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91影视

Simplify. $$\left(\frac{m^{4}}{m^{7}}\right)^{4}$$

Short Answer

Step by step solution

Simplify the Inner Fraction

Apply the Outer Exponent

Simplify the Final Expression

Key Concepts

Division of Exponents

Negative Exponents

Raising Powers to Powers

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