Problem 59 - 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 \(b\). $$ 12^{t} $$

Step-by-Step Solution

Answer

The expression \(12^t\) is equivalent to \(a \cdot b\).

Step 1: Break Down the Expression

To express \(12^t\) in terms of \(a\) and \(b\), start by breaking down the number 12 in terms of its prime factors. We know that \(12 = 2^2 \times 3\). This allows us to rewrite the expression as \((2^2 \cdot 3)^t\).

Step 2: Apply Power of a Product

Using the power of a product property \((x\cdot y)^n = x^n \cdot y^n\), rewrite \((2^2 \cdot 3)^t\) as \((2^2)^t \cdot 3^t\).

Step 3: Simplify Each Term

Simplify each individual term in the expression. For \((2^2)^t\), use the power of a power property \((x^m)^n = x^{m\cdot n}\) to get \(2^{2t}\). Therefore, \( 2^{2t} = (2^t)^2 = a^2\).

Step 4: Express 3^t in terms of a and b

We want to express \(3^t\) in terms of \(a\) and \(b\). We know \(6^t = (2 \times 3)^t = 2^t \cdot 3^t\). From this, we have \(b = a \cdot 3^t\). Solving for \(3^t\), we get \(3^t = \frac{b}{a}\).

Step 5: Combine Both Parts

Substitute the expressions from Step 3 and Step 4 into the original expression \((2^2)^t \cdot 3^t\). This gives us: \(a^2 \cdot \frac{b}{a}\).

Step 6: Simplify the Expression

Simplify \(a^2 \cdot \frac{b}{a}\) by canceling out one \(a\) from the numerator and denominator. This leads to the final expression \(a \cdot b\).

Key Concepts

Power of a Product

When dealing with exponentiation in mathematical expressions, the power of a product rule is essential. This rule states that for any numbers \(x\) and \(y\), and an exponent \(n\), \((x \cdot y)^n = x^n \cdot y^n\). This means that you can distribute the exponent to each factor inside the parentheses.

Imagine you have an expression like \((2 \cdot 3)^t\). Using the power of a product, it turns into \(2^t \cdot 3^t\). This property helps in breaking down complex expressions into parts that are easier to manage.

• Make sure each factor within the product is raised to the power separately.
• It simplifies handling products within exponents.
• Understanding this rule is key in algebraic manipulations involving exponents.

Power of a Power

The power of a power rule is another useful law of exponents. It explains how to deal with exponents that are themselves raised to another exponent. For any number \(x\), and exponents \(m\) and \(n\), \((x^m)^n = x^{m \cdot n}\). Essentially, you multiply the exponents together.

Consider the expression \((2^2)^t\). Applying the power of a power rule, you simplify it to \(2^{2t}\). This reduces complexities by making calculations straightforward.

• Multiply exponents when one is raised by another.
• Simplifies nested exponent scenarios.
• Crucial for solving complex exponential expressions.

Prime Factorization

Prime factorization involves expressing a number as a product of its prime numbers. Prime numbers are integers greater than 1, which have no divisors other than 1 and themselves. This technique helps break down composite numbers into a form useful for algebraic operations.

For example, the number 12 can be expressed as \(2^2 \times 3\). Here, 2 and 3 are prime factors of 12. Using prime factorization, you transform numbers into their basic building blocks. This makes it easier to apply exponent rules straightforwardly.

• Breaks composite numbers down to make calculations easier.
• Facilitates the use of laws of exponents.
• Helps in identifying the structure needed for applying rules like power of a product.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form for easier interpretation and computation. This might involve canceling terms, applying exponent rules, or performing arithmetic operations. It typically results in a more manageable expression and is a fundamental goal in algebra.

For instance, take the expression \(a^2 \cdot \frac{b}{a}\) derived from using laws of exponents. By simplifying, you cancel out \(a\) in the numerator and the denominator, resulting in \(a \cdot b\). Achieving this shows the importance of simplifying for solving mathematical problems efficiently.

• Enhances clarity and computational ease.
• Involves using rules like distributing, factoring, or cancelling terms.
• Allows you to express results in terms of known variables like \(a\) and \(b\).