Problem 137 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ x^{3}-64-(x+4)\left(x^{2}+4 x-16\right) $$

Step-by-Step Solution

Answer

The statement \(5^{-2} > 2^{-5}\) is false. The correct statement is \(5^{-2} < 2^{-5}\).

Step 1: Understanding the Given Expression

The given expression is \(5^{-2} > 2^{-5}\). The properties of exponents state that any non-zero number to a negative exponent equals its reciprocal to the positive exponent. Thus, \(5^{-2}\) can be written as \(\frac{1}{5^2}\), and \(2^{-5}\) can be re-written as \(\frac{1}{2^5}\).

Step 2: Simplify Each Expression

Now simplify each expression. \(5^2\) is 25, and \(2^5\) is 32. So, the inequality in simplified form becomes: \(\frac{1}{25} > \frac{1}{32}\).

Step 3: Compare the Values

The two quantities are both positive, but \(\frac{1}{25}\) is smaller than \(\frac{1}{32}\) (since in fractions with 1 as the numerator, larger denominator means smaller value), therefore, the original inequality \(5^{-2} > 2^{-5}\) is false.

Step 4: Convert the False Statement into a True Statement

To make the inequality true, the inequality sign has to be reversed, hence you will get: \(5^{-2} < 2^{-5}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, when you see something like \(5^3\), it means you are multiplying 5 by itself three times: \(5 \times 5 \times 5\). This results in the value 125.
Exponents have several key properties:

• \(a^m \times a^n = a^{m+n}\), where you add the exponents if the bases are the same.
• \(\frac{a^m}{a^n} = a^{m-n}\), where you subtract exponents when dividing.
• \((a^m)^n = a^{mn}\), which allows you to multiply the exponents when an exponent is raised to another power.

Understanding these exponent rules is essential when dealing with more complex math concepts, such as handling inequalities with exponents.

Reciprocal

The concept of a reciprocal is fundamental in math. Simply put, the reciprocal of a number is 1 divided by that number.
For instance:

• The reciprocal of 5 is \(\frac{1}{5}\).
• The reciprocal of \(\frac{1}{5}\) is back to 5.

When you encounter negative exponents, you utilize reciprocals. A negative exponent indicates that instead of multiplying, you take the reciprocal of the base raised to the opposite positive exponent.
Hence, \(5^{-2}\) becomes \(\frac{1}{5^2}\), giving you the reciprocal of \(5^2\). This helps transform complex problems into simpler fraction comparisons.

Negative Exponents

Negative exponents might seem tricky at first, but they're not too hard to grasp once you get the concept. When you see a negative exponent in expression, like \(a^{-b}\), you should think of turning it into a fraction.
Here's the basic rule: \[a^{-b} = \frac{1}{a^b}\]\This means that instead of multiplying \(a\), you perform division with it, raising it to the positive version of the exponent.
For example, consider \(2^{-5}\). This translates to:

• \(\frac{1}{2^5} = \frac{1}{32}\).

This rule is crucial for evaluating expressions and inequalities involving negative exponents, as it simplifies the comparison.

Inequality Comparison

When comparing inequalities with fractions, remember to pay attention to the values of the fractions. Specifically, when working with reciprocals (or fractions with 1 in the numerator), the size of the denominator plays a significant role.
For such fractions:

• A larger denominator means a smaller overall value.
• Conversely, a smaller denominator means a larger value.

Let's apply this to our original inequality. When given \(\frac{1}{25}\) and \(\frac{1}{32}\), although you might initially think that the numbers 25 and 32 are large, this context flips the interpretation.
The fraction \(\frac{1}{25}\) is smaller than \(\frac{1}{32}\) because 25 is less than 32.
Thus, the original expression \(5^{-2} > 2^{-5}\) is false, and reversing the inequality creates the true statement: \(5^{-2} < 2^{-5}\). Understanding these dynamics is crucial for accurately comparisons in math.