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Zorluk: OrtaOperations with Variable Exponents

Consider the algebraic expression below, where xx and yy represent non-zero values:

(4x3y−2)22x−1y3 \frac{(4x^3 y^{-2})^2}{2x^{-1} y^3}

Which of the following represents the completely simplified form of this expression?

  1. A
    8x5y7 \frac{8x^5}{y^7}
  2. 8x7y7 \frac{8x^7}{y^7} Cevap
  3. C
    2x7y7 \frac{2x^7}{y^7}
  4. D
    8x6y3 \frac{8x^6}{y^3}

Cevap

The completely simplified form is 8x7y7 \frac{8x^7}{y^7} .
The expression is simplified by first distributing the outer exponent of 2 to every factor in the numerator, which squares the coefficient 4 and multiplies the variables' exponents to produce 16x6y−416x^6y^{-4}. Next, dividing the coefficients yields 16÷2=816 \div 2 = 8. Then, the quotient rule is applied by subtracting the denominator's exponents from the numerator's exponents for matching bases: for xx, 6−(−1)=76 - (-1) = 7, and for yy, −4−3=−7-4 - 3 = -7. This results in 8x7y−78x^7y^{-7}. Finally, the negative exponent is resolved by moving y7y^7 to the denominator, giving the completely simplified form 8x7y7 \frac{8x^7}{y^7} .

Adım Adım Çözüm

1
Apply the power of a product rule to simplify the numerator.
(4x3y−2)2=42⋅(x3)2⋅(y−2)2=16x6y−4 (4x^3 y^{-2})^2 = 4^2 \cdot (x^3)^2 \cdot (y^{-2})^2 = 16x^6y^{-4}
When an algebraic term is raised to a power, the outer exponent multiplies the exponent of each factor inside the parentheses, including the numerical coefficient (which has an implied exponent of 1).
2
Rewrite the fraction using the expanded numerator.
16x6y−42x−1y3 \frac{16x^6y^{-4}}{2x^{-1} y^3}
This sets up the expression to cleanly apply the quotient rules for exponents and to divide the numerical coefficients.
3
Divide the numerical coefficients.
16÷2=8 16 \div 2 = 8
Coefficients follow standard arithmetic division rules independent of the variables.
4
Apply the quotient rule to the variables by subtracting the denominator's exponents from the numerator's exponents.
For xx: 6−(−1)=6+1=7 6 - (-1) = 6 + 1 = 7 . For yy: −4−3=−7 -4 - 3 = -7 .
The quotient rule states that aman=am−n \frac{a^m}{a^n} = a^{m-n} . Note that subtracting a negative number is algebraically equivalent to addition.
5
Combine the terms and convert the negative exponent into a positive exponent.
8x7y−7=8x7y7 8x^7y^{-7} = \frac{8x^7}{y^7}
A negative exponent represents the reciprocal of the base (a−n=1an a^{-n} = \frac{1}{a^n} ). Moving y−7y^{-7} to the denominator as y7y^7 yields the final simplified expression format.

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Operations with Variable Exponents
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Operations with Variable Exponents

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