Problem 61 - Precalculus with Calculus Previews

Question

Assume that \(2^{t}=a\) and \(6^{t}=b\). Use the laws of exponents given in this section to express the value of the given expression in terms of \(a\) and \(b\). $$ 6^{-t} $$

Step-by-Step Solution

Answer

\(6^{-t} = \frac{1}{b}\)

Step 1: Rewrite the Expression

Given the expression is \(6^{-t}\). According to exponential laws, \(6^t\) is equivalent to \(b\), where \(b = 6^t\). Hence, \(6^{-t}\) can be rewritten based on its relation to \(b\).

Step 2: Use the Reciprocal Property

We know that \(6^{-t} = \frac{1}{6^t}\). Since \(6^t = b\), then \(\frac{1}{6^t} = \frac{1}{b}\). Therefore, \(6^{-t} = \frac{1}{b}\).

Step 3: Verify with Rules of Exponents

Recall the rules of exponents which state that \(a^{-n} = \frac{1}{a^n}\). Our expression is in the correct form: \(6^{-t} = \frac{1}{b}\). This confirms our solution.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical phrases involving exponents, where a number is repeatedly multiplied by itself. This number is called the base, and the exponent indicates how many times to multiply the base. For example, in the expression \(2^t\), 2 is the base, and \(t\) is the exponent. To visualize exponential growth, think about the expression \(2^t\): if \(t = 1\), then \(2^t = 2\); for \(t = 2\), \(2^t = 2 \times 2 = 4\); and similarly for larger values of \(t\). Exponential expressions are a key component of various fields like math and science because they succinctly represent large numbers and rapid growth patterns. When dealing with these expressions, it's crucial to understand how to manipulate them using exponent rules, such as when expressing in terms like \(a = 2^t\) and \(b = 6^t\), which can then be used to simplify complex problems as seen in the exercise.

Reciprocal Property

The reciprocal property is a fundamental concept when dealing with negative exponents. When you encounter a negative exponent, like in the expression \(6^{-t}\), it signifies the reciprocal of that base raised to the positive of the exponent. Put simply, \(a^{-n} = \frac{1}{a^n}\). So for \(6^{-t}\), you can rewrite it as \(\frac{1}{6^t}\). Using the earlier promise, \(6^t = b\), it quickly becomes clear that \(6^{-t} = \frac{1}{b}\). This property helps simplify expressions and solve equations because it breaks down complex parts into manageable chunks using basic division. By understanding the reciprocal property, you'll be better equipped to tackle various mathematical problems.

Rules of Exponents

Rules of exponents are the foundational principles that govern how exponents are handled in mathematical expressions. These rules include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)
• Zero Exponent Rule: \(a^0 = 1\) for any nonzero \(a\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

These rules make it possible to transform tricky exponential problems into simpler, more manageable tasks. For instance, verifying that \(6^{-t} = \frac{1}{b}\) relies directly on the negative exponent rule, which states \(a^{-n} = \frac{1}{a^n}\). Understanding and applying these rules is crucial, especially in recognizing patterns and simplifying terms to solve equations effectively.