15 Aug, 2024

· Mathematics

Which multiplication expression is equivalent to x+8/x^2÷2x+16/2x^2

Short Answer

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Long Explanation

Explanation

Equivalent Multiplication Expression

To find a multiplication expression equivalent to

\frac{x + 8}{x^2} \div \frac{2x + 16}{2x^2}

we first rewrite the division as multiplication by the reciprocal.

\frac{x + 8}{x^2} \cdot \frac{2x^2}{2x + 16}

Next, we simplify the expressions.

\frac{x + 8}{x^2} \cdot \frac{2x^2}{2(x + 8)}

Notice that $2x + 16$ is factored as $2(x + 8)$ .

Now we simplify:

\frac{1}{x} \cdot \frac{1}{1} = \frac{1}{x}

Thus, the equivalent multiplication expression is:

\frac{x + 8}{x^2} \cdot \frac{2x^2}{2(x + 8)} = \frac{1}{x}

Therefore, the multiplication expression equivalent to the given complex fraction is

\frac{1}{x}

Verified By

Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

mathematics

Concept

Simplification Of Rational Expressions

Explanation

Simplification of rational expressions involves reducing a rational expression to its simplest form. A rational expression is a fraction where both the numerator and the denominator are polynomials. The goal of simplification is to make this fraction as simple as possible.

Steps to Simplify Rational Expressions

Factor both the numerator and the denominator: Identify and factor out the greatest common factors (GCF) of the polynomials in both the numerator and the denominator.
Cancel out common factors: Any factor that appears in both the numerator and the denominator can be cancelled out.

Example

Consider the rational expression:

\frac{6x^2 + 12x}{3x}

Step 1: Factor the numerator and the denominator.

Numerator: $6x^2 + 12x = 6x(x + 2)$
Denominator: $3x = 3x$

So the expression becomes:

\frac{6x(x + 2)}{3x}

Step 2: Cancel out the common factors between the numerator and the denominator.

Since $6x$ and $3x$ share a common factor of $3x$ , we cancel it out:

\frac{6x(x + 2)}{3x} = \frac{2(x + 2)}{1} = 2(x + 2)

So, the simplified form is:

2(x + 2)

Important Notes

Only common factors can be cancelled out.
Ensure that the rational expression is defined (denominator ≠ 0) after simplifying.
Simplification does not change the value but makes it easier to understand or use in further calculations.

Common Mistakes

Leaving the factorization incomplete: Make sure to fully factorize both the numerator and the denominator.
Cancelling terms that are not factors: Only factors, not terms, can be cancelled. For example, in $\frac{x+2}{x+3}$ , $x$ cannot be cancelled as they are terms, not factors.

Understanding these fundamental steps and practicing multiple examples can significantly help in mastering the simplification of rational expressions.

Concept

Reciprocal Of A Fraction

Explanation

The reciprocal of a fraction is a new fraction obtained by swapping the numerator and the denominator of the original fraction.

If we have a fraction:

\frac{a}{b}

where $a$ is the numerator and $b$ is the denominator, its reciprocal will be:

\frac{b}{a}

Steps to Find the Reciprocal

Identify the numerator and the denominator of the fraction.
Interchange the positions of the numerator and the denominator.
The resulting fraction is the reciprocal.

Example

For the fraction:

\frac{3}{4}

its reciprocal will be:

\frac{4}{3}

Note

For a whole number, $n$ , its reciprocal is:

\frac{1}{n}

since a whole number can be thought of as a fraction with the denominator of 1.

Important Property

Multiplicative Inverse: The product of a fraction and its reciprocal is always 1.

\frac{a}{b} \times \frac{b}{a} = 1

Understanding reciprocal helps in solving equations involving fractions, finding multiplicative inverses, and simplifying complex fractions.