What is the value of [1/(1 + x b - a + x c - a ) + 1/(1 + x a - b + x c - b ) + 1/(1 + x a - c + x b - c )] where x ≠ 0?
1
Understanding the Algebraic Expression with Exponents
The question asks for the value of a specific algebraic expression involving exponents. The expression is:
\[ \frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{a - c} + x^{b - c}} \]We are given that \(x \ne 0\). To find the value of this expression, we need to simplify each term.
Step-by-Step Simplification of Each Term
Let's simplify each of the three fractions in the sum. We will use the basic exponent rule that states \(x^{m-n} = \frac{x^m}{x^n}\).
Simplifying the First Term
The first term is \( \frac{1}{1 + x^{b - a} + x^{c - a}} \). Let's rewrite the denominator using the exponent rule:
Denominator \( = 1 + x^{b - a} + x^{c - a} = 1 + \frac{x^b}{x^a} + \frac{x^c}{x^a} \)
To combine these terms into a single fraction, we find a common denominator, which is \(x^a\):
\[ 1 + \frac{x^b}{x^a} + \frac{x^c}{x^a} = \frac{x^a}{x^a} + \frac{x^b}{x^a} + \frac{x^c}{x^a} = \frac{x^a + x^b + x^c}{x^a} \]Now, substitute this back into the first term of the original expression:
\[ \text{First Term} = \frac{1}{\frac{x^a + x^b + x^c}{x^a}} \]When dividing by a fraction, we multiply by its reciprocal:
\[ \text{First Term} = 1 \cdot \frac{x^a}{x^a + x^b + x^c} = \frac{x^a}{x^a + x^b + x^c} \]Simplifying the Second Term
The second term is \( \frac{1}{1 + x^{a - b} + x^{c - b}} \). Let's simplify its denominator:
Denominator \( = 1 + x^{a - b} + x^{c - b} = 1 + \frac{x^a}{x^b} + \frac{x^c}{x^b} \)
The common denominator here is \(x^b\):
\[ 1 + \frac{x^a}{x^b} + \frac{x^c}{x^b} = \frac{x^b}{x^b} + \frac{x^a}{x^b} + \frac{x^c}{x^b} = \frac{x^b + x^a + x^c}{x^b} \]Substitute this back into the second term:
\[ \text{Second Term} = \frac{1}{\frac{x^b + x^a + x^c}{x^b}} = 1 \cdot \frac{x^b}{x^b + x^a + x^c} = \frac{x^b}{x^a + x^b + x^c} \]Note that the denominator \(x^b + x^a + x^c\) is the same as \(x^a + x^b + x^c\).
Simplifying the Third Term
The third term is \( \frac{1}{1 + x^{a - c} + x^{b - c}} \). Let's simplify its denominator:
Denominator \( = 1 + x^{a - c} + x^{b - c} = 1 + \frac{x^a}{x^c} + \frac{x^b}{x^c} \)
The common denominator here is \(x^c\):
\[ 1 + \frac{x^a}{x^c} + \frac{x^b}{x^c} = \frac{x^c}{x^c} + \frac{x^a}{x^c} + \frac{x^b}{x^c} = \frac{x^c + x^a + x^b}{x^c} \]Substitute this back into the third term:
\[ \text{Third Term} = \frac{1}{\frac{x^c + x^a + x^b}{x^c}} = 1 \cdot \frac{x^c}{x^c + x^a + x^b} = \frac{x^c}{x^a + x^b + x^c} \]Again, the denominator \(x^c + x^a + x^b\) is the same as \(x^a + x^b + x^c\).
Combining the Simplified Terms
Now we add the three simplified terms together:
\[ \text{Expression Value} = \frac{x^a}{x^a + x^b + x^c} + \frac{x^b}{x^a + x^b + x^c} + \frac{x^c}{x^a + x^b + x^c} \]Since all three fractions have the same denominator, \(x^a + x^b + x^c\), we can add the numerators directly:
\[ \text{Expression Value} = \frac{x^a + x^b + x^c}{x^a + x^b + x^c} \]Assuming that the denominator \(x^a + x^b + x^c\) is not equal to zero, the fraction simplifies to 1.
Therefore, the value of the given expression is 1.
Revision Table: Key Concepts in Simplification
| Concept | How it was Applied | Rule Used |
|---|---|---|
| Exponent Rule: \(x^{m-n}\) | Used to rewrite terms like \(x^{b-a}\) as \(x^b/x^a\). | \(x^{m-n} = \frac{x^m}{x^n}\) |
| Finding Common Denominator | Used to combine terms in the denominator of each fraction. | Allows adding fractions with different denominators after rewriting them. |
| Adding Fractions with Same Denominator | Used to sum the three simplified terms. | \( \frac{P}{Q} + \frac{R}{Q} + \frac{S}{Q} = \frac{P+R+S}{Q} \) |
| Simplifying a Fraction | Used when the numerator and denominator are identical. | \( \frac{A}{A} = 1 \) (where \(A \ne 0\)) |
Additional Information: Importance of Exponent Rules
Exponent rules are fundamental in algebra and are used extensively in simplifying expressions and solving equations. The rule \(x^{m-n} = \frac{x^m}{x^n}\) is particularly useful when dealing with differences in exponents.
- Remember that \(x^0 = 1\) for any \(x \ne 0\).
- Also, \(x^{-n} = \frac{1}{x^n}\). This is closely related to the rule \(x^{0-n} = \frac{x^0}{x^n} = \frac{1}{x^n}\).
- When simplifying expressions, always look for opportunities to use these rules to make the expression simpler.
- The condition \(x \ne 0\) is important because division by zero is undefined.
Practicing various problems involving exponent rules helps in mastering algebraic manipulations.
Similar Questions
What is px2 + qy2 + rz2 equal to ?
If 2b = a + c and y 2= xz, then what is x b - c yc - a za - b‑ equal to?
What is the value of [(√5 - √3) / (√5 + √3)] - [(√5 + √3) / (√5 - √3)]?
What is the square root of 23 - 4√15 ?
Which one of the following is the largest number among 2222 2, 222 22 , 22 222 , 2 2222 ?
If x = 2 + 2 2/3 + 2 1/3 , then the value of the expression x 3- 6x 2+ 6x will be
If a = xy p - 1 , b = yz q - 1 , c = zx r - 1 , then a q - r br - p cp - q is equal to
If x = y 1/a , y = z 1/b and z = x 1/c where x ≠ 1, y ≠ 1 and z ≠ 1, then what is the value of abc?
- If 4 x 2 y = 128 and 3 3x 3 2y − 9 xy = 0, then the value of x + y can be equal to
If \(\mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}}\) , then what is the value of m?
Important Questions from Surds and Indices
Find the cube root of 78402752
Find the value of :
[(3 × 3 × 3 × 3 × 3 × 3) 6 ÷ (3 × 3 × 3 × 3) 7 × 3 4]
The cube root of - 64 × - 1331 is:
If (27) m = (81) n, then m 2: mn = ?
if 49 n + 49 n + 49 n + 49 n + 49 n + 49 n + 49 n = 7 2221 , then n = ?
