If a = xy p - 1 , b = yz q - 1 , c = zx r - 1 , then a q - r br - p cp - q is equal to
None of the above
Understanding the Algebraic Expression Problem
The question asks us to simplify and find the value of the expression \( a^{q-r} b^{r-p} c^{p-q} \), given the definitions of \( a \), \( b \), and \( c \). The given definitions are:
- \( a = xy^{p-1} \)
- \( b = yz^{q-1} \)
- \( c = zx^{r-1} \)
We need to substitute these definitions into the expression and simplify it using the rules of exponents.
Step-by-Step Simplification of Powers and Exponents
Let's substitute the given values of \( a \), \( b \), and \( c \) into the expression \( a^{q-r} b^{r-p} c^{p-q} \):
\( a^{q-r} = (xy^{p-1})^{q-r} \)
\( b^{r-p} = (yz^{q-1})^{r-p} \)
\( c^{p-q} = (zx^{r-1})^{p-q} \)
Now, we use the exponent rule \( (uv)^m = u^m v^m \) and \( (u^n)^m = u^{nm} \):
\( a^{q-r} = x^{q-r} \cdot (y^{p-1})^{q-r} = x^{q-r} y^{(p-1)(q-r)} \)
\( b^{r-p} = y^{r-p} \cdot (z^{q-1})^{r-p} = y^{r-p} z^{(q-1)(r-p)} \)
\( c^{p-q} = z^{p-q} \cdot (x^{r-1})^{p-q} = z^{p-q} x^{(r-1)(p-q)} \)
Next, we multiply these simplified terms together:
\( a^{q-r} b^{r-p} c^{p-q} = (x^{q-r} y^{(p-1)(q-r)}) \cdot (y^{r-p} z^{(q-1)(r-p)}) \cdot (z^{p-q} x^{(r-1)(p-q)}) \)
Group the terms with the same base (x, y, and z):
\( = x^{q-r} x^{(r-1)(p-q)} \cdot y^{(p-1)(q-r)} y^{r-p} \cdot z^{(q-1)(r-p)} z^{p-q} \)
Now, use the exponent rule \( u^m \cdot u^n = u^{m+n} \) to combine the exponents for each base:
Exponent of x: \( (q-r) + (r-1)(p-q) \)
Exponent of y: \( (p-1)(q-r) + (r-p) \)
Exponent of z: \( (q-1)(r-p) + (p-q) \)
Calculating Exponents of x, y, and z
Let's simplify each exponent:
Exponent of x: \( (q-r) + (r-1)(p-q) \) \( = q - r + (rp - rq - p + q) \) \( = q - r + rp - rq - p + q \) \( = rp - rq - p + 2q - r \)
Exponent of y: \( (p-1)(q-r) + (r-p) \) \( = (pq - pr - q + r) + r - p \) \( = pq - pr - q + 2r - p \)
Exponent of z: \( (q-1)(r-p) + (p-q) \) \( = (qr - qp - r + p) + p - q \) \( = qr - qp - r + 2p - q \)
Final Result and Comparison with Options
Substituting these exponents back into the expression, we get:
\( a^{q-r} b^{r-p} c^{p-q} = x^{rp - rq - p + 2q - r} y^{pq - pr - q + 2r - p} z^{qr - qp - r + 2p - q} \)
Let's compare this result with the given options:
- Option 1: \( abc = (xy^{p-1})(yz^{q-1})(zx^{r-1}) = x \cdot x^{r-1} \cdot y^{p-1} \cdot y \cdot z^{q-1} \cdot z = x^{1+r-1} y^{p-1+1} z^{q-1+1} = x^r y^p z^q \)
- Option 2: \( xyz = x^1 y^1 z^1 \)
- Option 3: \( 0 \)
The derived expression \( x^{rp - rq - p + 2q - r} y^{pq - pr - q + 2r - p} z^{qr - qp - r + 2p - q} \) does not generally simplify to \( x^r y^p z^q \), \( xyz \), or \( 0 \) for arbitrary values of \( p \), \( q \), and \( r \).
For example, if \( p=1, q=2, r=3 \), then \( a = x y^0 = x \), \( b = y z^1 = yz \), \( c = z x^2 = zx^2 \). The exponents of the expression \( a^{q-r} b^{r-p} c^{p-q} \) are \( q-r = 2-3 = -1 \), \( r-p = 3-1 = 2 \), \( p-q = 1-2 = -1 \). The expression is \( a^{-1} b^2 c^{-1} = (x)^{-1} (yz)^2 (zx^2)^{-1} = x^{-1} y^2 z^2 z^{-1} x^{-2} = x^{-1-2} y^2 z^{2-1} = x^{-3} y^2 z^1 \). Option 1: \( x^r y^p z^q = x^3 y^1 z^2 \). This is not equal to \( x^{-3} y^2 z^1 \). Option 2: \( xyz = x^1 y^1 z^1 \). This is not equal to \( x^{-3} y^2 z^1 \). Option 3: \( 0 \). Not equal.
Therefore, based on the step-by-step simplification, the result is not among the first three options.
| Term | Definition | Power in Expression | Expanded Form |
|---|---|---|---|
| \( a \) | \( xy^{p-1} \) | \( q-r \) | \( x^{q-r} y^{(p-1)(q-r)} \) |
| \( b \) | \( yz^{q-1} \) | \( r-p \) | \( y^{r-p} z^{(q-1)(r-p)} \) |
| \( c \) | \( zx^{r-1} \) | \( p-q \) | \( z^{p-q} x^{(r-1)(p-q)} \) |
The combined exponents for \( x \), \( y \), and \( z \) are complex expressions of \( p, q, r \) and do not universally reduce to simple integers like 1 or values that would match the options.
Conclusion on Algebraic Expression Value
Since the simplified expression \( x^{rp - rq - p + 2q - r} y^{pq - pr - q + 2r - p} z^{qr - qp - r + 2p - q} \) is not equal to \( abc \), \( xyz \), or \( 0 \), the correct answer is None of the above.
Revision Table: Exponents and Algebraic Simplification
| Exponent Rule | Description | Example |
|---|---|---|
| \( (u^n)^m = u^{nm} \) | Power of a power | \( (x^2)^3 = x^{2 \times 3} = x^6 \) |
| \( (uv)^m = u^m v^m \) | Power of a product | \( (xy)^3 = x^3 y^3 \) |
| \( u^m \cdot u^n = u^{m+n} \) | Product of powers with the same base | \( x^2 \cdot x^3 = x^{2+3} = x^5 \) |
Additional Information on Similar Algebraic Problems
Problems involving cyclic expressions like this often result in a simple value like 1, especially when the exponents sum to zero, as \( (q-r) + (r-p) + (p-q) = 0 \) in this case.
A common form that results in 1 is \( (x/y)^{p-1} (y/z)^{q-1} (z/x)^{r-1} \). If the question was formulated this way, let's see the result:
\( \left(\frac{x}{y}\right)^{p-1} \left(\frac{y}{z}\right)^{q-1} \left(\frac{z}{x}\right)^{r-1} \) \( = \frac{x^{p-1}}{y^{p-1}} \cdot \frac{y^{q-1}}{z^{q-1}} \cdot \frac{z^{r-1}}{x^{r-1}} \) \( = x^{p-1} \cdot y^{-(p-1)} \cdot y^{q-1} \cdot z^{-(q-1)} \cdot z^{r-1} \cdot x^{-(r-1)} \) \( = x^{p-1 - (r-1)} \cdot y^{-(p-1) + (q-1)} \cdot z^{-(q-1) + (r-1)} \) \( = x^{p-1 - r + 1} \cdot y^{-p + 1 + q - 1} \cdot z^{-q + 1 + r - 1} \) \( = x^{p-r} \cdot y^{q-p} \cdot z^{r-q} \)
This expression does not necessarily equal 1 unless \( p=q=r \). However, if the exponents of the bases were \( q-r \), \( r-p \), \( p-q \) respectively, like \( (x/y)^{q-r} (y/z)^{r-p} (z/x)^{p-q} \), then the exponent of x would be \( (q-r) - (p-q) = q-r-p+q = 2q-r-p \), which still doesn't universally simplify to 0.
Another common form resulting in 1 is \( x^{q-r} y^{r-p} z^{p-q} \). If \( a=x, b=y, c=z \), the expression would be this, and if \( p=q=r \), the original expression simplifies to \( x^0 y^0 z^0 = 1 \).
Given the provided options and the standard formats of such problems, it's likely that the intended result was something simple like 1 or xyz, and the complexity derived from the direct interpretation suggests "None of the above" is the correct choice for this specific problem statement.
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