/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 39

Simplify each expression. $$ \left(\frac{m^{10}}{n}\right)^{8} $$

Short Answer

Expert verified
The simplified expression is \( \frac{m^{80}}{n^8} \).

Step by step solution

01

Apply the Power of a Quotient Rule

We start by applying the power of a quotient rule, which states that \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). Using this rule, we transform \( \left( \frac{m^{10}}{n} \right)^{8} \) into \( \frac{(m^{10})^8}{n^8} \).
02

Apply the Power of a Power Rule

Next, we apply the power of a power rule, which states that \( (a^m)^n = a^{m \cdot n} \). Using this rule on \( (m^{10})^8 \), we get \( m^{80} \). Therefore, our expression is now \( \frac{m^{80}}{n^8} \).
03

Simplify the Expression

The expression \( \frac{m^{80}}{n^8} \) is already in its simplest form because there are no additional common bases or terms that can be simplified. Thus, the final simplified expression is \( \frac{m^{80}}{n^8} \).

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Key Concepts

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

Power of a Quotient Rule
When simplifying expressions like \( \left( \frac{m^{10}}{n} \right)^{8} \), the Power of a Quotient Rule is very helpful. This rule explains how to handle the situation when a fraction is raised to a power. It can be expressed as \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). In simpler terms:

  • Raise both the numerator and denominator of the fraction to the same power separately.
  • Only then you continue with other operations, since the math within the fraction is preserved.
A practical example of this is our expression, \( \left( \frac{m^{10}}{n} \right)^{8} \), which we broke down into \( \frac{(m^{10})^8}{n^8} \). This method allows us to handle each part of the fraction efficiently.
Power of a Power Rule
The Power of a Power Rule is useful when you need to simplify expressions where you have a power raised to another power. This rule can be represented as \( (a^m)^n = a^{m \cdot n} \). It implies that:

  • You multiply the exponents together, instead of adding them.
  • This process simplifies computation and makes large exponents more manageable.
In our example, after applying the Power of a Quotient Rule first, we faced \((m^{10})^8\). Using the Power of a Power Rule, it became \(m^{80}\). By multiplying \(10\) by \(8\), we directly calculated the new exponent, leading us closer to the final simplified expression.
Simplifying Expressions
With all the rules applied, reaching the final simplified expression is your ultimate goal. Simplifying expressions means you want to have an expression that is easy to understand and work with. This often implies:

  • Ensuring no further operations or factorizable terms are left unresolved.
  • The expression should be as short and as clear as possible.
In our case, after applying the Power of a Power and Power of a Quotient rules, we arrived at the expression \( \frac{m^{80}}{n^8} \). There are no like terms to combine or simplify further, so this result is considered the simplest form. Always verify if the expression can be simplified further, ensuring optimum clarity and efficiency in your calculations.

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Most popular questions from this chapter

Construct a table of values for each polynomial function using the given values for \(x .\) Then graph the function and find its domain and range. $$ \begin{aligned} &f(x)=-x^{3}-x^{2}+6 x\\\ &x=-4,-3,-2,-1,0,1,2,3 \end{aligned} $$ $$ \begin{array}{|r|r|} \hline x & f(x) \\ \hline-4 & \\ -3 & \\ -2 & \\ -1 & \\ 0 & \\ 1 & \\ 2 & 3 & \\ \hline \end{array} $$

$$ \begin{aligned} &f(x)=2 x^{2}-4 x+2\\\ &x=-1,0,1,2,3 \end{aligned} $$ $$ \begin{array}{|r|r|} \hline x & f(x) \\ \hline-1 & \\ 0 & \\ 1 & \\ 2 & \\ 3 & \\ \hline \end{array} $$

Factor each expression. $$ (a+b) x^{3}+27(a+b) $$

Solve each inequality. Graph the solution set and write it using interval notation. $$ |2 x+1| \geq 7 $$

The surface area of a cubical block of ice is represented by the polynomial \(6 x^{2}+36 x+54 .\) Use factoring to find an expression that represents the length of an edge of the block.
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