Simplify -7/(-4+ square root of 5)
The problem is asking to perform the simplification of a rational expression, which is given as the fraction -7 divided by the sum of -4 and the square root of 5. The expression inside the denominator involves the square root of 5 which is an irrational number, and a negative integer -4. Simplification in this context usually means to rationalize the denominator (if possible) which involves eliminating the square root from the denominator, and then simplify the resulting expression to its simplest form.
$\frac{- 7}{- 4 + \sqrt{5}}$
Answer
Solution:
Step 1:
Rewrite the expression by placing the negative sign outside the fraction. $- \frac{7}{-4 + \sqrt{5}}$
Step 2:
Rationalize the denominator by multiplying the fraction by $\frac{-4 - \sqrt{5}}{-4 - \sqrt{5}}$. $- \left( \frac{7}{-4 + \sqrt{5}} \cdot \frac{-4 - \sqrt{5}}{-4 - \sqrt{5}} \right)$
Step 3:
Perform the multiplication in the numerator. $- \frac{7(-4 - \sqrt{5})}{(-4 + \sqrt{5})(-4 - \sqrt{5})}$
Step 4:
Apply the difference of squares formula to the denominator. $- \frac{7(-4 - \sqrt{5})}{16 - (\sqrt{5})^2}$
Step 5:
Simplify the denominator by evaluating $(\sqrt{5})^2$ and simplifying the expression. $- \frac{7(-4 - \sqrt{5})}{11}$
Step 6:
Distribute the negative sign in the numerator. $- \frac{7(-1(4) - \sqrt{5})}{11}$
Step 7:
Factor out the negative one from the terms in the numerator. $- \frac{7(-1(4 + \sqrt{5}))}{11}$
Step 8:
Simplify the expression by removing the double negative. $- (- \frac{7(4 + \sqrt{5})}{11})$
Step 9:
Final simplification steps.
Step 9.1:
Cancel out the negative signs. $- (- \frac{7(4 + \sqrt{5})}{11})$
Step 9.2:
Simplify to a single negative sign. $1 \cdot \frac{7(4 + \sqrt{5})}{11}$
Step 9.3:
The final result is obtained by multiplying by 1. $\frac{7(4 + \sqrt{5})}{11}$
Step 10:
Present the final result in various forms.
Exact Form: $\frac{7(4 + \sqrt{5})}{11}$
Decimal Form: Approximately $3.96840689 \ldots$
Knowledge Notes:
To simplify the given expression, we use the following knowledge points:
Rationalizing the Denominator: When a fraction has a radical (square root) in the denominator, we can eliminate the radical by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of $a + b$ is $a - b$.
Difference of Squares: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. This is used to simplify the denominator after rationalizing it.
Simplifying Square Roots: The square of a square root eliminates the radical, so $(\sqrt{a})^2 = a$.
Distributive Property: This property allows us to multiply a single term by each term inside a parenthesis, such as $a(b + c) = ab + ac$.
Negative Signs: A negative sign in front of a fraction can be distributed into the numerator or denominator. Additionally, the product of two negative signs is positive.
Exact vs. Decimal Form: The exact form of a number includes radicals and fractions, while the decimal form is an approximation of the number in decimal notation.
By applying these concepts, we can simplify the given expression step by step, ensuring that each transformation maintains the equality of the expression.
