Problem 7 - Precalculus: Mathematics for Calculus

Question

Yes or No? If No, give a reason. (a) Is the expression \(\left(\frac{2}{3}\right)^{-2}\) equal to \(\frac{3}{4} ?\) (b) Is there a difference between \((-5)^{4}\) and \(-5^{4} ?\)

Step-by-Step Solution

Answer

(a) No, because \(\left(\frac{2}{3}\right)^{-2} = \frac{9}{4}\), not \(\frac{3}{4}\). (b) Yes, the values are different (625 and -625).

Step 1: Simplify the First Expression

The expression given is \(\left(\frac{2}{3}\right)^{-2}\). To simplify this expression, we need to use the rule for negative exponents which states that \(a^{-b} = \frac{1}{a^b}\). Therefore, \(\left(\frac{2}{3}\right)^{-2} = \frac{1}{\left(\frac{2}{3}\right)^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4}\).

Step 2: Compare with Given Expression

We have simplified \(\left(\frac{2}{3}\right)^{-2}\) to \(\frac{9}{4}\). The question asks if this is equal to \(\frac{3}{4}\). Since \(\frac{9}{4} eq \frac{3}{4}\), the expression \(\left(\frac{2}{3}\right)^{-2}\) is not equal to \(\frac{3}{4}\).

Step 3: Analyze Negatively and Positively Paired Base

The second part asks about \((-5)^4\) and \(-5^4\). \((-5)^4\) means we are raising the negative number \(-5\) to the power 4, which results in \((-5) \times (-5) \times (-5) \times (-5) = 625\). \(-5^4\) means we raise 5 to the power 4 and then apply the negative sign: \(-1 \times 5^4 = -625\).

Step 4: Compare Results of Second Expression

From Step 3, we can see that \((-5)^4 = 625\), while \(-5^4 = -625\). Therefore, \((-5)^4\) and \(-5^4\) are not the same.

Key Concepts

Negative Exponents

Negative exponents might seem tricky, but they are straightforward once you grasp the basic rule. When you encounter a negative exponent, it essentially means you're dealing with the reciprocal of the base raised to the corresponding positive exponent.
For instance, if you have a base like \(b^{-n}\), according to the rule, this becomes \(\frac{1}{b^{n}}\). By flipping the base and changing the sign of the exponent, you can transform complex negative-exponent problems into simple positive ones.

• This rule is essential because it simplifies expressions and makes calculations easier.
• It also aids in understanding how negative exponents influence the size of numbers, showing a form of inverse growth compared to positive exponents.

Order of Operations

Understanding the order of operations is crucial in mathematics to solve expressions correctly. A common tool used to remember this order is the acronym 'PEMDAS', which stands for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
This order tells us which operations to perform first.

• It's particularly relevant when dealing with expressions that include multiple operations to ensure that calculations are performed in the right sequence.
• Mistakes in following the correct order can lead to incorrect results, as seen in example (b) where misunderstanding this rule can produce different outcomes.

Ensuring you stick to this order makes the difference between a right and a wrong answer.

Exponent Notation

Exponent notation is an efficient way to express repeated multiplication of a number by itself. If you have \(a^b\), 'a' is the base, and 'b' is the exponent, indicating that a is multiplied by itself b times.
This notation simplifies the representation of large numbers and simplifies calculations in algebra and beyond.

• It clarifies expressions, making them more manageable, especially in polynomial expressions.
• It's useful in scientific contexts where large numbers frequently come into play, enabling easier computation.

Understanding how different placements of negative signs and parentheses affect this notation is crucial, as seen in the difference between expressions such as \((-5)^4\) and \(-5^4\).

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical expressions into their simplest form. This can include reducing fractions, combining like terms, and using rules like those for exponents.
For example, when you encounter \(\left(\frac{2}{3}\right)^{-2}\), simplification involves applying the rule of negative exponents, resulting in \frac{9}{4}\..

• This process makes it much easier to handle the expression in practical real-life applications or when solving equations.
• Simplification helps in identifying equivalent expressions or in recognizing patterns, making complex problems more manageable.

Through simplifying, you can verify whether different expressions represent the same value, like determining that \frac{9}{4}\ is not equal to \frac{3}{4}\.