Chapter 5: Problem 34
\(\frac{2}{\sqrt{8}+\sqrt{7}}\)
Short Answer
Expert verified
\( 2\sqrt{8}-2\sqrt{7} \)
Step by step solution
01
Multiply by Conjugate
In order to eliminate the square roots from the denominator, multiply the numerator and the denominator by the conjugate of the denominator: \( \frac{2}{\sqrt{8}+\sqrt{7}} \times \frac{\sqrt{8}-\sqrt{7}}{\sqrt{8}-\sqrt{7}} \)
02
Foil in the Denominator
Use the FOIL (First, Outer, Inner, Last) method in the denominator: \( \sqrt{8}\times\sqrt{8} - \sqrt{8}\times\sqrt{7} + \sqrt{7}\times\sqrt{8} - \sqrt{7}\times\sqrt{7} \)
03
Simplification of the Denominator
Simplify the denominator: \( 8 - \sqrt{56} + \sqrt{56} - 7 \)
04
Simplification of the Numerator
Simplify the numerator: \( 2(\sqrt{8}-\sqrt{7}) \)
05
Final simplification
Simplify the final equation: \( \frac{2\sqrt{8}-2\sqrt{7}}{1} \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate of a Binomial
Understanding the conjugate of a binomial is essential when simplifying radical expressions, particularly those that involve square roots. A binomial conjugate refers to a pair of binomial expressions where both have the same terms, but the sign between them is opposite. For example, if we have a binomial \(a+b\), its conjugate is \(a-b\).
The reason conjugates are useful is that when you multiply a binomial by its conjugate, you get a difference of squares. This means that the radical parts of the expressions eliminate each other, leaving you with an expression that no longer has square roots. In the exercise, multiplying the expression by the conjugate \(\sqrt{8}-\sqrt{7}\) makes it possible to simplify the denominator without dealing with messy square roots.
The reason conjugates are useful is that when you multiply a binomial by its conjugate, you get a difference of squares. This means that the radical parts of the expressions eliminate each other, leaving you with an expression that no longer has square roots. In the exercise, multiplying the expression by the conjugate \(\sqrt{8}-\sqrt{7}\) makes it possible to simplify the denominator without dealing with messy square roots.
FOIL Method
The FOIL method is an acronym that stands for First, Outer, Inner, Last, and it provides a systematic way to multiply two binomials. To FOIL effectively, you take the first term in each binomial and multiply them together (\textbf{First}). Next, multiply the outer terms (\textbf{Outer}), followed by the inner terms (\textbf{Inner}), and lastly, the last terms in each binomial are multiplied (\textbf{Last}).
This method streamlines the process of expanding binomials. In our exercise, applying the FOIL method to \(\sqrt{8}+\sqrt{7}\) times its conjugate \(\sqrt{8}-\sqrt{7}\) helps to simplify the expression systematically. It is especially powerful because it transforms the denominator into a rational number, paving the way for the cancellation of the square roots.
This method streamlines the process of expanding binomials. In our exercise, applying the FOIL method to \(\sqrt{8}+\sqrt{7}\) times its conjugate \(\sqrt{8}-\sqrt{7}\) helps to simplify the expression systematically. It is especially powerful because it transforms the denominator into a rational number, paving the way for the cancellation of the square roots.
Square Roots
The concept of square roots is foundational in simplifying radical expressions. A square root, symbolized by \(\sqrt{}\), represents a number that, when multiplied by itself, gives the original number. For instance, \(\sqrt{9}\) is 3 because \(3 \times 3 = 9\).
In the step-by-step solution, we handle square roots within a complex fraction. By simplifying square roots like \(\sqrt{8}\), which is equivalent to \(\sqrt{4 \times 2}\) or \(2\sqrt{2}\), we can express them in their simplest radical form. Simplifying square roots is key because it often leads to further simplifications or the ability to combine like terms, as seen in steps 3 and 5 of our solution.
In the step-by-step solution, we handle square roots within a complex fraction. By simplifying square roots like \(\sqrt{8}\), which is equivalent to \(\sqrt{4 \times 2}\) or \(2\sqrt{2}\), we can express them in their simplest radical form. Simplifying square roots is key because it often leads to further simplifications or the ability to combine like terms, as seen in steps 3 and 5 of our solution.
Algebraic Expressions
An algebraic expression is a mathematical statement that includes numbers, variables, and arithmetic operations (such as addition and multiplication), and sometimes exponents and roots. Simplifying an algebraic expression involves combining like terms and performing arithmetic operations to condense the expression into its simplest form.
In our example, we simplify the algebraic expression resulting from the multiplication of the fraction by its conjugate. We perform operations such as addition and subtraction with radicals and rational numbers to reach a more straightforward expression, \(2\sqrt{8}-2\sqrt{7}\), where we no longer have a radical in the denominator, making it easier to comprehend and further manipulate if needed.
In our example, we simplify the algebraic expression resulting from the multiplication of the fraction by its conjugate. We perform operations such as addition and subtraction with radicals and rational numbers to reach a more straightforward expression, \(2\sqrt{8}-2\sqrt{7}\), where we no longer have a radical in the denominator, making it easier to comprehend and further manipulate if needed.
