Evaluate each expression for p=−4p=-4, q=8q=8, and r=−10r=-10. 3p+3(4+p)3r+8\frac{3p + 3(4 + $$p)^3$$}{r + 8} r+83p+3(4+p)3

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 111

1장, 문제 111

Evaluate each expression for $p = - 4$ , $q equals 8$ , and $r = - 10$ . $\frac{3p + 3(4 + p)^3}{r + 8}$

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First, substitute the given values into the expression: replace $ p $ with $ -4 $, $ q $ with $ 8 $ (though $ q $ is not used in this expression), and $ r $ with $ -10 $. The expression becomes $ 3(-4) + \frac{3(4 + (-4))^3}{-10 + 8} $.

Simplify inside the parentheses: calculate $ 4 + (-4) $ which simplifies to $ 0 $.

Evaluate the exponent: raise the result from step 2 to the power of 3, so calculate $ 0^3 $.

Calculate the numerator and denominator separately: multiply $ 3 $ by $ p $ (which is $ -4 $) for the first term, and multiply $ 3 $ by the result of the exponentiation for the numerator of the fraction. Then simplify the denominator by adding $ r $ and $ 8 $.

Finally, combine the terms by adding the first term and the fraction, and simplify the expression to get the final value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Order of Operations

The order of operations dictates the sequence in which parts of an expression are evaluated: parentheses first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction. This ensures consistent and correct evaluation of expressions.

Substitution of Variables

Substitution involves replacing variables in an expression with given numerical values. This step is essential to evaluate algebraic expressions for specific values, allowing the expression to be simplified to a numerical result.

Exponentiation

Exponentiation is the operation of raising a base number to a power, indicating repeated multiplication. Understanding how to correctly compute powers, especially with negative bases or sums inside parentheses, is crucial for accurate evaluation.

Evaluate each expression for p=−4p=-4, q=8q=8, and r=−10r=-10. 3p+3(4+p)3r+8\(\frac{3p + 3(4 + p)^3}{r + 8}\) r+83p+3(4+p)3