An engineering formula used to calculate a structural load index is given by L = \frac{a^3… — ACT

An engineering formula used to calculate a structural load index is given by $L = \frac{a^3 b^{-2} - c^{3/4}}{a^2 + 12b}$ . What is the value of $L$ when $a = -2$ , $b = \frac{1}{3}$ , and $c = 16$ ?

Cevap: -10

Cevap

The value of the expression is -10.

Substituting the values into the formula gives

L = \frac{(-2)^3 \cdot (1/3)^{-2} - 16^{3/4}}{(-2)^2 + 12(1/3)} = \frac{-8 \cdot 9 - 8}{4 + 4} = \frac{-72 - 8}{8} = \frac{-80}{8} = -10

Adım Adım Çözüm

Evaluate the terms in the numerator containing powers and negative exponents

a^3 = -8

b^{-2} = 9

, and

c^{3/4} = 8

Negative bases raised to odd powers remain negative:

(-2)^3 = -8

. A negative exponent represents the reciprocal raised to a positive exponent:

(1/3)^{-2} = 3^2 = 9

. Fractional exponent

c^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8

Compute the full numerator

-72 - 8 = -80

Multiply

a^3

and

b^{-2}

to get

(-8)(9) = -72

, then subtract

c^{3/4} = 8

Evaluate the denominator

(-2)^2 + 12\left(\frac{1}{3}\right) = 4 + 4 = 8

Squaring a negative number yields a positive value:

(-2)^2 = 4

. Multiplying

12 \cdot \frac{1}{3} = 4

Divide the numerator by the denominator

\frac{-80}{8} = -10

Dividing a negative integer by a positive integer yields a negative result.

Anahtar Kavram

Evaluating algebraic expressions involving negative numbers, negative exponents, and rational exponents

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