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Chapter 1: Problem 78

Evaluate. $$ \frac{a^{3}-4 a}{a(a-3)}, \text { for } a=-2 $$

Short Answer

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Step by step solution

01

Substitute the given value of a

We are given the expression \[ \frac{a^3 - 4a}{a(a-3)} \] and the value of \( a = -2 \). Substitute \( a = -2 \) into the expression.
02

Calculate the numerator

Substitute \( a = -2 \) into the numerator which is \( a^3 - 4a \). \[ (-2)^3 - 4(-2) = -8 + 8 = 0 \].
03

Calculate the denominator

Substitute \( a = -2 \) into the denominator which is \( a(a-3) \). \[ -2(-2-3) = -2(-5) = 10 \].
04

Divide the numerator by the denominator

With the numerator being 0 and the denominator being 10, \[ \frac{0}{10} = 0 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method involves replacing variables in an algebraic expression with their given values. This simplifies the expression, making it easier to evaluate. In the exercise, you were given the expression \( \frac{a^3 - 4a}{a(a-3)} \) and asked to evaluate it for \( a = -2 \).
To do this:
  • Substitute \( a = -2 \) into all instances of \( a \) in the expression.
  • This substitution reduces the algebraic expression to numerical calculations, which are much easier to handle.
Numerator and Denominator
In a fraction, the numerator is the top part and the denominator is the bottom part. For example, in \[ \frac{a^3 - 4a}{a(a-3)} \], \[ a^3 - 4a \] is the numerator, and \[ a(a-3) \] is the denominator. This distinction is important because each part must be calculated separately before performing the division.
When evaluating an expression where both parts are algebraic, you substitute the given value of the variable into the numerator and the denominator individually. Then you perform any necessary arithmetic operations.
Here, after substitution:
  • The numerator was calculated as: \( (-2)^3 - 4(-2) = -8 + 8 = 0 \)
  • The denominator was calculated as: \( -2(-2-3) = -2(-5) = 10 \)
Once the calculations are done, you proceed to the next step of the evaluation.
Division in Algebra
Division in algebra involves dividing the simplified forms of the numerator and denominator. In the given exercise, after we substituted \( a = -2 \) and simplified, we found the numerator to be 0 and the denominator to be 10.
It鈥檚 important to remember a few key points about division in algebra:
  • If the numerator is 0, the entire fraction will be 0, provided the denominator is not zero.
  • If the denominator is 0, the division is undefined as you cannot divide by zero.
  • In this case: the final division was \( \frac{0}{10} = 0 \), since zero divided by any non-zero number is always zero.
Understanding these rules helps simplify and accurately solve algebraic expressions involving division.

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Most popular questions from this chapter

Evaluate. $$ \frac{3 a-4 a^{2}}{a^{2}-20}, \text { for } a=5 $$

Simplify using a calculator. Round your answer to the nearest thousandth. $$ \frac{46-(3-8)^{3}}{2\left[35-(18-26)^{2}\right]} $$

Tell whether each statement is true or false for all real numbers m and n. Use various replacements for m and n to support your answer. If \(m=-n,\) then \(m+n=0\)

Evaluate. $$ \frac{x^{2}+2^{x}}{x^{2}-2^{x}}, \text { for } x=3 $$

SYNTHESIS Is it true that for all real numbers \(a\) and \(b\) $$ a b=(-a)(-b) ? $$ Why or why not?
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