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Chapter 1: Problem 9

\(6 x^{-7}-4 \sqrt{x}\)

Short Answer

Expert verified
The simplified form of \(6 x^{-7}-4 \sqrt{x}\) is \(\frac{6}{x^7} - 4\sqrt{x}\)

Step by step solution

01

Break It Down

Given the expression \(6 x^{-7}-4 \sqrt{x}\), we can split this into two parts: the term with the negative exponent \((6 x^{-7})\) and the term with the square root \((- 4 \sqrt{x})\).
02

Handle the Negative Exponent

In mathematics, \(x^{-n}\) denotes the reciprocal of \(x^n\). Therefore, \(x^{-7}\) can be written as \(\frac{1}{x^7}\). So, the term \(6 x^{-7}\) can be rewritten as \(\frac{6}{x^7}\).
03

Handle the Square Root

\(\sqrt{x}\) means the value that is multiplied by itself to give \(x\). Here, this value is just multiplied by -4, so we can鈥檛 simplify it any further.
04

Put It Back Together

Now, combine the terms from steps 2 and 3 to form the simplified algebraic expression: \(\frac{6}{x^7} - 4\sqrt{x}\).

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Key Concepts

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

Negative Exponents
Negative exponents can seem tricky at first, but they're more straightforward than they appear. When you see an expression like \(x^{-n}\), it means you're dealing with the reciprocal of \(x^n\). In simpler terms, it鈥檚 as though you were flipping \(x^n\) upside down. This helps turn a negative power into a positive one.

Here's a quick way to handle it:
  • \(x^{-n} = \frac{1}{x^n}\)
In our example, \(x^{-7}\) becomes \(\frac{1}{x^7}\). This change from a negative exponent to a fraction is vital for simplifying expressions.
Remembering the reciprocal connection allows you to confidently manage negative exponents in any algebra problem.
Square Roots
Square roots are fundamental in mathematics. The symbol \(\sqrt{}\) represents a number that, when multiplied by itself, gives the original number inside the square root.

For instance:
  • \(\sqrt{x}\)
means "what number multiplied by itself equals \(x\)". In many expressions, especially simple ones, you might not need further simplification unless you know \(x\).
In the expression \(-4\sqrt{x}\), the square root of \(x\) is simply multiplied by \(-4\), which means unless you have more information about \(x\), this is as simplified as it gets.
Understanding square roots helps make sense of equations with them and how they interact with other terms.
Reciprocal in Mathematics
The concept of a reciprocal is tied closely to the idea of flipping a fraction. When you take a number and find its reciprocal, you're effectively trading places between the numerator and the denominator.

Consider a number \(a\):
  • The reciprocal of \(a\) is \(\frac{1}{a}\).
This becomes especially useful when dealing with expressions like negative exponents. When \(x^{-n}\) turns into \(\frac{1}{x^n}\), you鈥檙e leveraging the reciprocal concept.
This simplification technique helps you manage equations and fractions neatly.
Understanding reciprocals not only aids with simplifying expressions but also plays a crucial role in operations involving division and multiplication in algebra.

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Most popular questions from this chapter

The acceleration of a particle moving along the \(x\) -axis at time \(t\) is given by \(a(t)=4 t-12 .\) If the velocity is 10 when \(t=0\) and the position is 4 when \(t=0\) , then the particle is changing direction at (A) \(t=1\) (B) \(t=3\) (C) \(t=5\) (D) \(t=1\) and \(t=5\)

If \(\frac{d y}{d x}=\frac{y^{2}}{x^{3}}\) and \(y(1)=2,\) find an equation for \(y\) in terms of \(x\)

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