Examine These Problems; Find The Mistake In Each Show

Examine The Following Problems Find The Mistake In Each Show That Th

Examine the following problems. Find the mistake in each. Show that the problems are incorrect by substituting the following values for the variables: a = 3, b = 2, n = 4; or by working both sides of the equation and showing that they are not equal, then rework the problems correctly, explaining your steps and using Equation Editor.

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The assignment requires evaluating a series of algebraic problems to identify and correct errors. The process involves substituting specific values for variables and verifying the correctness of each equation. If discrepancies are found, the correct solution is to be derived with clear explanations. This exercise emphasizes understanding algebraic manipulations, simplification techniques, and the importance of precise mathematical notation.

Let's analyze each problem carefully, substitute the given values (a = 3, b = 2, n = 4), and determine whether the statements are true or contain mistakes. For problems where mistakes are evident, we will rework the calculations to find the correct solutions.

Problem 1

Original statement: ????????3 · ????????4 = ???????? = 54

Interpreting symbols, let us assume the problem is: a3 · a4 = a = 54

Substitute a = 3:

  • Left side: 33 · 34 = (27) · (81) = 2187
  • Right side: 54

Since 2187 ≠ 54, the equation is incorrect. The correct evaluation of the left side is 2187, not 54.

Proper correction involves recognizing the property of exponents:

a3 · a4 = a3+4 = a7

Substituting a = 3, we get 37 = 2187. Therefore, the correct statement is:

33 · 34 = 37 = 2187

Problem 3

Original statement: (???????? + ????????)2 = ????????2 + ????????????????

Assuming ???????? = a, ???????? = b, the equation reads:

(a + b)2 = a2 + b2

Substitute a = 3, b = 2:

  • Left side: (3 + 2)2 = 52 = 25
  • Right side: 32 + 22 = 9 + 4 = 13

Since 25 ≠ 13, the equation is incorrect. The correct expansion of (a + b)2 is:

(a + b)2 = a2 + 2ab + b2

For a = 3 and b = 2:

9 + 2(3)(2) + 4 = 9 + 12 + 4 = 25

which matches the left side. The mistake was neglecting the middle term 2ab.

Problem 4

Original statement: ??????????2 + ??????????3 = 4????????

Assuming ???????? = a, ???????? = b, the equation reads:

a2 + b3 = 4a (or possibly 4b). But since the variable is the same on both sides, let's clarify.

Given the structure, perhaps the equation is: a2 + b3 = 4a

Substitute a = 3, b = 2:

  • Left side: 32 + 23 = 9 + 8 = 17
  • Right side: 4 × 3 = 12

Since 17 ≠ 12, the equation is incorrect. The corrected form might be:

a2 + b3 = 4a is false for these values. Alternatively, if the original intended was different, clarification is essential.

Problem 6

Original statement: ????????−???????? = ????????????????

Assuming ???????? = a, ???????? = b:

a − b = ? (unknown)

Substitute a = 3, b = 2:

3 − 2 = 1

So, a − b = 1, which suggests the right side is 1, but the original statement is incomplete. If corrected:

a − b = 1

This is correct with the substitution.

Problem 7

Original statement: (???????? + ????????)−1 = 1 ???????? + 1 ????????

Interpreting as inverse of sum: (a + b)−1 = 1/a + 1/b

Let's verify with a=3, b=2:

Left side: (3 + 2)−1 = 1/5 = 0.2

Right side: 1/3 + 1/2 ≈ 0.333 + 0.5 = 0.833

Since 0.2 ≠ 0.833, the equality is false. The correct relation is:

(a + b)−1 = 1/a + 1/b ?

Actually, 1 / (a + b) ≠ 1/a + 1/b; instead:

1 / (a + b) = {straightforwardly; no simple relation to 1/a + 1/b}

The mistake is assuming inverse distributivity over addition, which is incorrect.

Problem 8

Original statement: (2????????2)3 = 6????????5

Interpreted as: (2a2)3 = 6a5

Substitute a=3:

  • Left: (2 · 32)3 = (2 · 9)3 = (18)3 = 5832
  • Right: 6 · 35 = 6 · 243 = 1458

Since 5832 ≠ 1458, the original statement is false. Let’s check the exponents:

Left side: (2a2)3 = 23 · a6 = 8a6

Right side: 6a5

Comparing: 8a6 vs 6a5

Substitute a=3:

  • Left: 8 · 36 = 8 · 729 = 5832
  • Right: 6 · 35 = 6 · 243 = 1458

Thus, the original expression is inconsistent; the actual correct simplification shows the exponents differ.

Conclusion

This exercise underscores the importance of understanding exponent rules, algebraic properties, and correct application of formulas. Many of the original equations presented contain typographical or conceptual errors, such as missing middle terms, incorrect distributive assumptions, or misunderstood properties of exponents and reciprocals. Through substitution and explicit calculation, we confirm the inaccuracies and recommend the corrected forms as outlined above.

References

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