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Chapter 10: Problem 307

$$\frac{48 x^{6}}{6 x^{4}}-\frac{35 x^{9}}{7 x^{7}}$$

Short Answer

Expert verified
\( 3 x^{2} \)

Step by step solution

01

Simplify the First Fraction

Simplify the fraction \( \frac{48 x^{6}}{6 x^{4}} \). Divide the coefficients: \( \frac{48}{6} = 8 \). Subtract the exponents of the same base: \( x^{6-4} = x^{2} \). Therefore, \( \frac{48 x^{6}}{6 x^{4}} = 8 x^{2} \).
02

Simplify the Second Fraction

Simplify the fraction \( \frac{35 x^{9}}{7 x^{7}} \). Divide the coefficients: \( \frac{35}{7} = 5 \). Subtract the exponents of the same base: \( x^{9-7} = x^{2} \). Therefore, \( \frac{35 x^{9}}{7 x^{7}} = 5 x^{2} \).
03

Subtract the Results

From the previous steps, we have \( 8 x^{2} \) and \( 5 x^{2} \). Subtract these: \( 8 x^{2} - 5 x^{2} = 3 x^{2} \).

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Key Concepts

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

Fraction Simplification
When simplifying algebraic fractions, start by breaking down the problem.
A fraction consists of a numerator and a denominator. In algebraic fractions, both can include variables and exponents.
To simplify, follow these steps:
  • **Divide the coefficients:** This is like simplifying a basic arithmetic fraction. For example, in the fraction \( \frac{48}{6} \), divide \( 48 \) by \( 6 \) to get \( 8 \).
  • **Subtract exponents:** With similar bases, subtract the exponent in the denominator from the exponent in the numerator. For example, \( x^{6} \text{ and } x^{4} \), subtract four from six to get two, resulting in \( x^{2} \).
Combining these results simplifies the fraction.
For instance, \( \frac{48 x^{6}}{6 x^{4}} \) simplifies to \( 8 x^{2} \).
By simplifying each fraction in the expression, you'll make subtraction much easier!
Exponent Rules
Understanding the rules of exponents is crucial.
One primary rule is: **To divide with the same base, subtract the exponents**.
This means, if you have \( \frac{x^{a}}{x^{b}} \), it simplifies to \( x^{a-b} \).
  • **Example 1:** \( \frac{x^{6}}{x^{4}} = x^{6-4} = x^{2} \)
  • **Example 2:** \( \frac{x^{9}}{x^{7}} = x^{9-7} = x^{2} \)
Another essential rule is: **Any number raised to the power of zero is one**.
This means \( x^{0} = 1 \), an important concept for simplifying expressions.
Always keep these rules handy to simplify complex algebraic fractions easily!
Algebraic Subtraction
After simplifying fractions, the next step is to subtract them.
For example, from our previous steps, we have: \( 8 x^{2} \) and \( 5 x^{2} \).
  • **Align like terms:** To subtract, ensure the variables and exponents are the same. In this case, both terms are \( x^{2} \), so we can subtract directly.
  • **Subtract the coefficients:** Subtract the second coefficient from the first. For \( 8 x^{2} - 5 x^{2} \), this results in \( 3 x^{2} \).
Thus, the final answer is \( 3 x^{2} \), showcasing an algebraic subtraction of like terms.
Always double-check your simplified terms and perform accurate subtraction!

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Most popular questions from this chapter

In the following exercises, write each number in scientific notation. $$0.0000103$$

Simplify each expression. $$p^{-3} \cdot p^{-8}$$

Convert each number to decimal form. $$1.5 \times 10^{8}$$

Factor the greatest common factor from each polynomial. $$15 r+35$$

In the following exercises, simplify each expression with exponents. $$(0.4)^{3}$$
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