Problem 63 - 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 \(b\). $$ 2^{-3 t} 2^{7 t} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a^4\).

Step 1: Simplify the Expression Using Exponent Laws

The expression \(2^{-3t} \times 2^{7t}\) can be simplified using the property of exponents which states that \(a^m \times a^n = a^{m+n}\). Applying this, we have:\[2^{-3t + 7t} = 2^{(7t - 3t)} = 2^{4t}.\]

Step 2: Express in Terms of Given Variables

We know that \(2^t = a\). Therefore, any power of \(2^t\) can be rewritten in terms of \(a\). Since \(2^{4t} = (2^t)^4\), then:\[2^{4t} = a^4.\]

Key Concepts

Understanding Exponents

Exponents are a shorthand way to represent repeated multiplication of the same number. For example, in the expression \(2^5\), the base number \(2\) is multiplied by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\). This is especially helpful in mathematics because it allows complex calculations to be expressed more simply.
When working with exponents, there are specific laws or properties that make simplifying expressions possible. A key property to remember is that when you multiply two exponents with the same base, such as \(a^m \times a^n\), you can add the exponents, resulting in \(a^{m+n}\). This rule is a cornerstone of operations involving exponents and is crucial for simplifying expressions like \(2^{-3t} \times 2^{7t}\). Understanding this rule will help you manipulate and simplify exponential expressions efficiently.

Simplifying Expressions with Exponents

Simplifying an expression means making it as straightforward as possible by reducing it to its simplest form. When expressions involve exponents, the laws of exponents become incredibly handy.
Consider the expression \(2^{-3t} \times 2^{7t}\). To simplify this, you employ the exponent law that allows you to add the exponents when you multiply terms with the same base, so you calculate \((-3t + 7t)\), which results in \(2^{4t}\).
This simplification occurs because when you multiply powers of the same base, you simply add their exponents. Another important aspect is recognizing that exponents can be negative, which means the term is actually a fraction (i.e., \(2^{-x} = \frac{1}{2^x}\)). By applying these principles, you can significantly reduce the complexity of exponential expressions in your calculations.

Substituting Variables in Exponential Expressions

Variable substitution helps in rewriting expressions to reveal their form or to connect them to other variables. In our case, you're given that \(2^t = a\) and need to express \(2^{4t}\) in terms of \(a\).
Understanding substitution involves recognizing how a variable is defined. If \(2^t = a\), then a power of this, like \(2^{4t}\), can be rewritten by recognizing \(4t\) as \(t \times 4\), thus \((2^t)^4 = a^4\). This substitution means every occurrence of \(2^t\) in your calculation is replaced with \(a\).
Substituting with variables allows for flexibly recasting expressions into forms that may be preferable or that match other given conditions in a problem, making the expression easier to assess and work with.