Problem 9 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(2^{-3}=\frac{1}{2} \quad\) b. \(\frac{1}{t^{-6}}=t\) c. A factor can be moved from the denominator to the numerator or from the numerator to the denominator of a fraction if the _____ of its exponent is changed. d. A fraction raised to a power is equal to the ____ the fraction raised to the opposite power.

Step-by-Step Solution

Answer

a. 8; b. \(t^6\); c. sign; d. reciprocal.

Step 1: Understand the Problem

To solve this problem, we need to fill in the blanks with the appropriate mathematical terms or expressions. There are four parts to this exercise, each requiring a different kind of transformation or property of exponents.

Step 2: Solve Part a

We are given the expression \(2^{-3}\) and its equivalence which should be a fraction. Recall that a negative exponent means "take the reciprocal." Therefore, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\). Let's compare this to \(\frac{1}{2}\). To make the equal sign correct, it should say \(2^{-3} = \frac{1}{8}\), therefore the blank should be \(8\).

Step 3: Solve Part b

For the expression \(\frac{1}{t^{-6}}\), recognize that a negative exponent in the denominator should be moved to the numerator by changing the sign of the exponent. Thus, \(\frac{1}{t^{-6}} = t^6\). Therefore, the blank should be \(t^6\).

Step 4: Solve Part c

When a factor is moved from the denominator to the numerator or vice versa, the sign of its exponent changes. Therefore, the statement is: A factor can be moved from the denominator to the numerator or from the numerator to the denominator of a fraction if the **sign** of its exponent is changed.

Step 5: Solve Part d

A fraction raised to a power follows the property that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Therefore, a fraction raised to a power is equal to the reciprocal of the fraction raised to the opposite power when adjusting for certain transformations.

Key Concepts

Negative Exponents

Negative exponents can sometimes be a bit confusing at first, but they're actually quite simple once you get the hang of them. When you see a negative exponent, it indicates that you need to take the reciprocal of the base.

• For instance, if you have \( x^{-n} \), this is equivalent to \( \frac{1}{x^n} \). This means you essentially flip the base and make the exponent positive.
• Let's look at an example, if you have \( 5^{-2} \), this would simplify to \( \frac{1}{5^2} = \frac{1}{25} \).

Understanding that negative exponents simply "flip" the base is a powerful tool in simplifying expressions and solving equations.

By mastering this concept, you can solve more complex mathematical problems with ease.

Reciprocal

The concept of a reciprocal is closely tied to negative exponents. Simply put, the reciprocal of a number or expression is "what you multiply that number by to get 1."

• If you have a number \( x \), its reciprocal is \( \frac{1}{x} \).
• In the context of fractions, the reciprocal is achieved by flipping the fraction. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

When combined with the concept of exponents, particularly negative exponents, understanding reciprocals becomes very useful.

For example, \( x^{-1} \) represents the reciprocal of \( x \), which is \( \frac{1}{x} \). This direct relationship between negative exponents and reciprocals is essential for simplifying mathematical expressions effectively.

Exponent Properties

Exponent properties are rules that help us simplify and manipulate expressions with exponents. Here are some crucial properties and how they work:

• Product of Powers: \( x^a \times x^b = x^{a+b} \). This property allows us to add exponents when multiplying bases of the same number.
• Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \). This means when dividing like bases, you subtract the exponents.
• Power of a Power: \( (x^a)^b = x^{a \cdot b} \). You multiply the exponents when raising a power to another power.
• Power of a Product: \( (xy)^a = x^a \times y^a \). When you have a product raised to a power, you apply the exponent to each factor.
• Power of a Fraction: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). Each part of the fraction is raised to the power separately.

These properties are vital for simplifying expressions and solving equations involving exponents. Knowing them makes it easier to handle expressions that appear complicated at first.