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Zorluk: OrtaOperations with Variable Exponents

In a structural modeling calculation, an engineer must simplify a given formula before evaluating it. Which of the following expressions is mathematically equivalent to the algebraic fraction below?

16a5b2(2ab−1)3\frac{16a^5b^2}{(2ab^{-1})^3}
  1. 2a2b52a^2b^5Cevap
  2. B
    8a2b58a^2b^5
  3. C
    2a2b−12a^2b^{-1}
  4. D
    8a4b58a^4b^5

Cevap

The mathematically equivalent expression is 2a2b52a^2b^5.
The correct answer is derived by first applying the power rule to the entire denominator, yielding 8a3b−38a^3b^{-3}. Next, dividing the numerical coefficients (16÷8=216 \div 8 = 2) and applying the quotient rule for exponents (subtracting the denominator's exponents from the numerator's exponents) results in a5−3=a2a^{5-3} = a^2 and b2−(−3)=b2+3=b5b^{2-(-3)} = b^{2+3} = b^5. Combining these distinct parts yields 2a2b52a^2b^5.

Adım Adım Çözüm

1
Apply the power of a product rule to the denominator.
(2ab−1)3=23⋅a3⋅(b−1)3(2ab^{-1})^3 = 2^3 \cdot a^3 \cdot (b^{-1})^3
An exponent outside parentheses must be applied to every distinct factor inside the parentheses.
2
Simplify the factors in the denominator.
8a3b−38a^3b^{-3}
23=82^3 = 8, and for a power of a power (b−1)3(b^{-1})^3, the exponents are multiplied (−1×3=−3-1 \times 3 = -3).
3
Rewrite the fraction using the expanded denominator.
16a5b28a3b−3\frac{16a^5b^2}{8a^3b^{-3}}
This sets up the expression for application of the quotient rule.
4
Divide the numerical coefficients.
16÷8=216 \div 8 = 2
Constants are separated from variables and divided normally.
5
Apply the quotient rule to the variables with the base aa.
a5−3=a2a^{5-3} = a^2
When dividing like bases, subtract the exponent of the denominator from the exponent of the numerator.
6
Apply the quotient rule to the variables with the base bb.
b2−(−3)=b2+3=b5b^{2 - (-3)} = b^{2 + 3} = b^5
Subtracting a negative exponent is equivalent to adding a positive number.
7
Combine the simplified coefficient and variables.
2a2b52a^2b^5
Bringing the distinct evaluated parts back together provides the final simplified term.

Anahtar Kavram

To simplify a rational expression involving exponents, first use the power rule to eliminate grouping symbols, ensuring numerical coefficients are raised to the power. Then, divide the coefficients and use the quotient rule to subtract the denominator's exponents from the numerator's exponents for matching bases.
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Konular

Operations with Variable Exponents

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