Problem 93 Simplify. Do not use negative ex... [FREE SOLUTION]

Chapter 5: Problem 93

Simplify. Do not use negative exponents in the answer. \(\left(2 u^{-2} v^{5}\right)^{5}\)

Short Answer

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\(\frac{32 v^{25}}{u^{10}}\)

Step by step solution

Apply the Power of a Power Rule

To simplify the expression \(\left(2 u^{-2} v^{5}\right)^{5}\), begin by applying the power of a power rule. According to this rule, when an expression in the form \((a^m)^n\) is raised to a power, you multiply the exponents: \(a^{m \times n}\). Thus, we apply it to each component inside the parentheses: \(2^5 (u^{-2})^5 (v^5)^5\).

Simplify Each Component

Calculate each component separately: 1. For \(2^5\): Compute the power of 2 to the 5th, giving \(32\).2. For \((u^{-2})^5\): Multiply the exponents to get \(u^{-10}\).3. For \((v^5)^5\): Again, multiply the exponents to get \(v^{25}\). Thus, the intermediate expression becomes \(32 u^{-10} v^{25}\).

Eliminate Negative Exponents

Rewrite the expression to eliminate any negative exponents. Recall that \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\). Therefore, \(u^{-10}\) becomes \(\frac{1}{u^{10}}\). This transforms our expression into \(32 v^{25} \frac{1}{u^{10}}\) or equivalently, \(\frac{32 v^{25}}{u^{10}}\).

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Key Concepts

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

Power of a Power Rule

When dealing with exponents, one essential rule is the "power of a power" rule. Imagine you have an expression like \((a^m)^n\). Here, you're multiplying the exponent \(m\) by \(n\). This creates a new expression \(a^{m \times n}\). This concept is vital for simplifying expressions raised to another power.
In the exercise \((2 u^{-2} v^{5})^{5}\), each of the terms within the parentheses needs to be raised to the 5th power:

\(2^5\)
\((u^{-2})^5\)
\((v^5)^5\)

By applying the "power of a power" rule, we simplify each term individually. Let's see how this works in practice.

Negative Exponents

Negative exponents can be a bit tricky, but they follow a simple rule. When you see a negative exponent, \(a^{-n}\), it means you need to take the reciprocal of the base: \(\frac{1}{a^n}\). This rule transforms negative exponents into positive ones, making expressions easier to work with.
For instance, in \((u^{-2})^5\), we use the power of a power rule first, which results in \(u^{-10}\).
To eliminate any negative exponent, convert \(u^{-10}\) into a positive exponent by rewriting it as \(\frac{1}{u^{10}}\). This makes sure that all exponents in your final answer are positive, as required in the exercise.

Simplifying Expressions

When simplifying expressions, our goal is to express them in the most straightforward form. This often involves applying rules like the power of a power and converting negative exponents to positive ones. Let's look at our expression and see how it simplifies fully.
We start with \(2^5 (u^{-2})^5 (v^5)^5\) and simplify:

\(2^5 = 32\)
\((u^{-2})^5 = u^{-10} = \frac{1}{u^{10}}\)
\((v^5)^5 = v^{25}\)

After simplifying each portion, we combine them to form \(32 v^{25} \times \frac{1}{u^{10}}\).
This can be neatly written as a fraction: \(\frac{32 v^{25}}{u^{10}}\). By following each step carefully, from applying exponent rules to rewriting negative exponents, we've simplified the expression without using any negative exponents.

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91影视

Simplify. Do not use negative exponents in the answer. \(\left(2 u^{-2} v^{5}\right)^{5}\)

Short Answer

Step by step solution

Apply the Power of a Power Rule

Simplify Each Component

Eliminate Negative Exponents

Key Concepts

Power of a Power Rule

Negative Exponents

Simplifying Expressions

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