Examples of fraction in the following topics:
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- A complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.
- A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions.
- From previous sections, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction.
- You'll find that the common denominator of the two fractions in the numerator is 6, and then you can add those two terms together to get a single fraction term in the larger fraction's numerator:
- Recall, again, that dividing by a fraction is the same as multiplying by the reciprocal of that fraction:
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- A fraction represents a part of a whole.
- Find a common denominator, and change each fraction to an equivalent fraction using that common denominator.
- To subtract a fraction from a whole number or to subtract a whole number from a fraction, rewrite the whole number as a fraction and then follow the above process for subtracting fractions.
- To multiply a fraction by a whole number, simply multiply that number by the numerator of the fraction:
- The process for dividing a number by a fraction entails multiplying the number by the fraction's reciprocal.
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- Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.
- In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator.
- This same principal can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa.
- You are given the fraction $\frac{10}{\sqrt{3}}$, and you want to simplify it by eliminating the radical from the denominator.
- Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot
\frac{\sqrt{3}}{\sqrt{3}}
= {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$
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- In algebra, partial fraction decomposition (sometimes called partial fraction expansion) is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.
- We can then write $R(x)$ as the sum of partial fractions:
- We have rewritten the initial rational function in terms of partial fractions.
- We have solved for each constant and have our partial fraction expansion:
- There are some important cases to note, for which partial fraction decomposition becomes more complicated.
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- Then you rewrite the two fractions with this denominator.
- The denominator in the second fraction can not be factored.
- The first fraction has two factors: $y$ and $(x^2+2)$.
- The second fraction has one factor: $(x^2 + 2)$.
- We then rewrite both fractions with the common denominator.
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- A rational expression can be treated like a fraction, and can be manipulated via multiplication and division.
- A rational expression is a fraction involving polynomials, where the polynomial in the denominator is not zero.
- Just like a fraction involving numbers, a rational expression can be simplified, multiplied, and divided.
- Rational expressions can be multiplied and divided in a similar manner to fractions.
- Dividing rational expressions follows the same rules as dividing fractions.
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- Numbers with negative exponents are treated normally in arithmetic operations and can be rewritten as fractions.
- There is an additional rule that allows us to change the negative exponent to a positive one in the denominator of a fraction, and it holds true for any real numbers $n$ and $b$, where $b \neq 0$:
- To understand how this rule is derived, consider the following fraction:
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- In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
- Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations.
- Multiplying fractions a/b times c/d gives (ac)/(bd), whereas if one wanted to add a/b plus c/d, first convert them into ad/bd and cb/db, giving (ad+cb)/(db).
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- In such cases, the exponent acts as both a whole number exponent and a root, or fraction exponent.
- where $b$ is a real number and the rational exponent $\frac{m}{n}$ is a fraction in lowest terms.
- For example, we can rewrite $\sqrt{\frac{13}{9}}$ as a fraction with two radicals:
- In some cases, writing an exponent in its fraction form makes it easier to cancel powers and roots.
- We can simplify the fraction in the exponent to 2, giving us $5^2=25$.