Perform the following operation and express in simplest form.
start fraction, x, squared, minus, 64, divided by, x, squared, minus, 9, x, end fraction, dot, start fraction, x, plus, 9, divided by, x, squared, plus, x, minus, 72, end fraction
x
2
−9x
x
2
−64
⋅
x
2
+x−72
x+9
To simplify, we can factor the numerator and denominator separately.
Numerator:
x^2 - 64 can be factored as (x + 8)(x - 8).
Denominator:
x^2 - 9x can be factored as x(x - 9).
The expression becomes:
((x + 8)(x - 8))/(x(x - 9)) * (x + 9)/(x^2 + x - 72)
Next, we can cancel out common factors:
(x + 8)(x - 8)/(x(x - 9)) * (x + 9)/((x - 8)(x + 9))
The (x + 8) terms and (x + 9) terms cancel out:
1/(x(x - 9))
Therefore, the expression simplifies to 1/(x(x - 9)).
To solve this expression, we can follow these steps:
Step 1: Factor the numerator and denominator of both fractions.
The numerator of the first fraction is the difference of squares:
\(x^2 - 64 = (x + 8)(x - 8)\)
The denominator of the first fraction can be factored as a difference of cubes:
\(x^2 - 9x = x(x - 9)\)
The numerator of the second fraction cannot be factored further.
The denominator of the second fraction can be factored as a quadratic:
\(x^2 + x - 72 = (x + 9)(x - 8)\)
Step 2: Simplify the expression.
Now, we have:
\(\frac{(x + 8)(x - 8)}{x(x - 9)} \cdot \frac{x + 9}{(x + 9)(x - 8)}\)
Cancel out the common factors:
\(\frac{1}{x}\)
Therefore, the expression simplifies to \( \frac{1}{x} \).
To perform the given operation and express the result in its simplest form, follow these steps:
Step 1: Evaluate the numerator expression.
- Start by factoring the numerator expression: x^2 - 64
This can be factored as (x + 8)(x - 8)
- Now replace the numerator expression with the factored form: (x + 8)(x - 8)
Step 2: Evaluate the denominator expression.
- Factor the denominator expression: x^2 + x - 72
This can be factored as (x + 9)(x - 8)
- Replace the denominator expression with the factored form: (x + 9)(x - 8)
Step 3: Simplify the expression by canceling common factors.
- Cancel out the common factor (x - 8) in both the numerator and denominator.
- After canceling, the expression becomes:
(x + 8)/(x + 9)
Step 4: Express the simplified form of the expression.
- The final simplified form of the expression is:
(x + 8)/(x + 9)