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Chapter 3: Problem 101

Simplify the expression. $$\frac{3^{7 / 6}}{3^{1 / 6}}$$

Short Answer

Expert verified
The simplified expression is \(3^1\), which is simply 3.

Step by step solution

01

Identify the rule

There is a rule in exponents which states that \(a^m / a^n = a^{m-n}\). This rule applies when you divide terms with the same base. In this case, 3 is the common base.
02

Apply the rule

Applying the subtraction rule of exponents:\n\[\frac{3^{7 / 6}}{3^{1 / 6}} = 3^{(7 / 6)-(1 / 6)}\]
03

Simplify the expression

Subtracting the exponents:\n\[3^{(7 / 6)-(1 / 6)} = 3^{6 / 6} = 3^1\]

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is an essential skill in algebra that involves reducing expressions to their simplest form. This helps us deal with fewer terms and less complicated mathematics. When we simplify, we aim to express a complex expression in a way that is easier to understand and use.

In our example, we have the expression \( \frac{3^{7/6}}{3^{1/6}} \). Simplification often involves using different rules of exponents, like combining like terms or dealing with division and multiplication of powers. By following the procedures in rules of exponents, we make the expression simpler and more straightforward to evaluate.
Division of Exponents
When it comes to dealing with exponents, the division rule is another handy tool. This rule tells us how to divide powers with the same base, which is crucial in reducing expressions efficiently. If we have a fraction such as \( \frac{a^m}{a^n} \), it's equivalent to \( a^{m-n} \). This rule only applies when the bases are the same.

For our specific problem, the bases of both terms, \( 3^{7/6} \) and \( 3^{1/6} \), are both 3. This allows us to use the division rule to simplify the expression to a lower power of the base, making further calculations much simpler.
Subtracting Exponents
Subtracting exponents is a key process when using the division rule of exponents. It helps us condense expressions with multiple powers into a single power. By subtracting the exponents, we are effectively computing the power difference within the division.

In the given exercise, after identifying the common base, we subtract the exponents: \( 3^{(7 / 6)-(1 / 6)} \). This gives us the exponent \( 6/6 \), and when simplified, it becomes \( 3^1 \). Subtracting exponents simplifies the expression considerably, reducing complexity and making it easier to manage.
Common Base in Exponents
Using a common base is central to applying the rules of division and subtraction of exponents. When the base is the same in both the numerator and the denominator, it allows us to simplify the expression easily using these rules.

For this exercise, the base 3 is common in both parts of the expression \( \frac{3^{7/6}}{3^{1/6}} \). This commonality enables the straightforward application of exponent rules, ultimately simplifying the full expression to \( 3^1 \). Recognizing a common base is crucial since it determines whether the division rule and other exponent rules can be appropriately applied.

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Most popular questions from this chapter

The cost \(C\) (in dollars) of supplying recycling bins to \(p \%\) of the population of a rural township is given by $$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$ (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to \(15 \%\) \(50 \%,\) and \(90 \%\) of the population. (c) According to the model, would it be possible to supply bins to \(100 \%\) of the population? Explain.

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function might indicate that there should be one. $$h(x)=\frac{6-2 x}{3-x}$$

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$h(x)=x^{5}-7 x^{4}+10 x^{3}+14 x^{2}-24 x$$

Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division \(\left(x^{n}-1\right) /(x-1) .\) Create a numerical example to test your formula. $$\text { (a) } \frac{x^{2}-1}{x-1}=$$ $$\text { (b) } \frac{x^{3}-1}{x-1}=$$ $$\text { (c) } \frac{x^{4}-1}{x-1}=$$

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-1$$
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