Question
(n) \( \frac{a^{3 x+2 y+z} \cdot a^{x+3 y-z}}{a^{4 x+5 y}} \)
Ask by Donnelly Ball. in South Africa
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is 1.
Solution
To simplify the expression
(a^(3x + 2y + z) · a^(x + 3y − z)) / a^(4x + 5y)
follow these steps:
1. Combine the exponents in the numerator using the rule a^m · a^n = a^(m+n):
Exponent in numerator = (3x + 2y + z) + (x + 3y − z)
= 3x + x + 2y + 3y + z − z
= 4x + 5y
2. Now, the expression becomes:
a^(4x + 5y) / a^(4x + 5y)
3. Apply the quotient rule for exponents a^m / a^n = a^(m − n):
a^(4x + 5y − (4x + 5y)) = a^0
4. Any nonzero number raised to the 0th power equals 1:
a^0 = 1
Thus, the simplified expression is 1.
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Simplify this solution Beyond the Answer
To simplify the expression \( \frac{a^{3x + 2y + z} \cdot a^{x + 3y - z}}{a^{4x + 5y}} \), we first combine the powers in the numerator. Using the property of exponents that states \( a^m \cdot a^n = a^{m + n} \), we can add the exponents: \[ a^{(3x + 2y + z) + (x + 3y - z)} = a^{(3x + x) + (2y + 3y) + (z - z)} = a^{4x + 5y} \] Now, our expression simplifies to: \[ \frac{a^{4x + 5y}}{a^{4x + 5y}} \] Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we have: \[ a^{(4x + 5y) - (4x + 5y)} = a^0 \] Since \( a^0 = 1 \) for any non-zero \( a \), the final result is: \[ 1 \]
