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Chapter 3: Problem 5

Simplify the expression. $$ \frac{-6 x^{5}}{3 x} $$

Short Answer

Expert verified
-2x^4

Step by step solution

01

- Simplify the Fraction

Divide the numerator by the denominator: \( \frac{-6x^5}{3x} \). Simplify the coefficients: \( \frac{-6}{3} = -2 \). So, the simplified fraction becomes \( \frac{-6x^5}{3x} = \frac{-2x^5}{x} \).
02

- Apply the Law of Exponents

Use the exponent rule \( \frac{x^a}{x^b} = x^{a-b} \). Here, \( x^5 \) divided by \( x \) is \( x^{5-1} = x^4 \). Substitute this back into the expression: \( \frac{-2x^5}{x} = -2x^4 \).

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Key Concepts

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

fraction
Fractions are an essential part of algebra, and understanding how to simplify them is crucial. A fraction consists of a numerator (the top part) and a denominator (the bottom part). In our original expression, \( \frac{-6x^5}{3x} \), the numerator is \(-6x^5\) and the denominator is \(3x\).

To simplify a fraction, you divide the numerator by the denominator. This means dividing both the constants (numbers) and the variables.

In this specific case, we first simplify the constants: \( \frac{-6}{3} \rightarrow -2 \). Next, for the variables, since they are both the same base (x), you divide the exponents. Simplifying the coefficients is the first step in fraction simplification.
exponent rules
Exponents play a key role in algebra. When simplifying expressions with exponents, certain rules apply. One fundamental rule is the quotient of powers: \(\frac{x^a}{x^b} = x^{a-b} \).

Let's apply this rule to our problem. We need to divide \(x^5\) by \(x\). According to the rule, subtract the exponent in the denominator from the exponent in the numerator: \( x^5 \div x^1 = x^{5-1} = x^4 \).

Hence, our expression \( \frac{-6x^5}{3x} \) simplifies to \( -2x^4 \) after applying the exponent rules.
coefficient simplification
Coefficient simplification involves simplifying the numerical part of terms. Coefficients are the numbers in front of variables.

In our example, the coefficients are \(-6\) and \(3\). To simplify the coefficients in a fraction, you divide them just like normal numbers. So, \( \frac{-6}{3} = -2 \).

Thus, the coefficient part of our fraction is simplified from \( \frac{-6}{3} \) to \(-2\), making our expression more manageable.

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Most popular questions from this chapter

Write the domain of the function in interval notation. $$ s(x)=\frac{1}{\sqrt{4 x^{2}+7 x-2}} $$

Determine if the statement is true or false. If \(c\) is a zero of a polynomial \(f(x)\), with degree \(n \geq 2\) then all other zeros of \(f(x)\) are zeros of \(\frac{f(x)}{x-c}\).

Suppose that \(y\) varies inversely as the cube of \(x\). If the value of \(x\) is decreased to \(\frac{1}{4}\) of its original value, what is the effect on \(y\) ?

Determine if the statement is true or false. Given \(f(x)=2 i x^{4}-(3+6 i) x^{3}+5 x^{2}+7,\) if \(a+b i\) is a zero of \(f(x)\), then \(a-b i\) must also be a zero.

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{5-x}-7 \geq 0 $$
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