Problem 36 - Precalculus: Functions and Graphs

Question

Simplify. $$(8 r)^{13}\left(2 r^{1 / 2}\right)$$

Step-by-Step Solution

Answer

\((2^{40})r^{27/2}\)

Step 1: Separate the inital expression

Start by considering the expression \((8r)^{13}\left(2r^{1/2}\right)\) and separate it into two portions: \((8r)^{13}\) and \(2r^{1/2}\).

Step 2: Apply the power of a product rule

For the first part \((8r)^{13}\), apply the power of a product rule, which states \((ab)^n = a^n \, b^n\). This results in \(8^{13} \, r^{13}\).

Step 3: Simplify each factor

Now, simplify the parts separately: \((8^{13} \, r^{13})\) becomes \(8^{13} \, r^{13}\). As \(8^{13}\) is a large number, we will keep it as is for now.

Step 4: Simplify the second portion of the product

Now consider the second part of the original expression: \(2r^{1/2}\). Note that \(2r^{1/2}\) is already simplified, with \(r^{1/2}\) representing the square root of \(r\).

Step 5: Combine the simplified expressions

Combine the expressions by multiplying them together. It is expressed as: \(8^{13} \, r^{13} \, (2 \, r^{1/2})\).

Step 6: Use the properties of exponents for multiplication

For the terms involving \(r\), use the property that \(r^a \, r^b = r^{a+b}\). Thus, \(r^{13} \, r^{1/2} = r^{13 + 1/2}\).

Step 7: Calculate the exponent

Sum the exponents: \(13 + \frac{1}{2} = 13.5\) or 13 and a half, thus resulting in \(r^{27/2}\).

Step 8: Combine final terms

The expression then becomes \(8^{13} \, \cdot 2 \, \cdot r^{27/2}\). Calculate and further simplify as follows: \((8^{13} \, \cdot 2) \, r^{27/2}\) = \(2 \cdot 8^{13}\, r^{27/2}\).

Step 9: Simplify with constant

Since calculating \(8^{13}\) is complex without a calculator, express the coefficient as \(2(2^3)^{13} = 2 \cdot 2^{39} = 2^{40}\).

Key Concepts

Power of a Product Rule

In mathematics, the power of a product rule is a fundamental concept that makes dealing with exponents much easier.
This rule states that when you have a product raised to an exponent, like \((ab)^n\), you can distribute the exponent to both factors:

• The expression becomes \(a^n \cdot b^n\).

Applying this to the given problem of \((8r)^{13}\), we distribute the exponent \(13\) over both \(8\) and \(r\):

• The result is \(8^{13} \cdot r^{13}\).

Notice how helpful this is because it breaks a complex expression into manageable parts.
Thus, any product inside an exponent can be treated separately, simplifying calculations significantly.
This approach aligns neatly with other rules for exponents, making this principle invaluable when working through algebraic expressions.

Properties of Exponents

Exponents have various properties that make simplifying expressions straightforward and intuitive. Two essential properties observed in this problem are:

• ;The power of a product rule (discussed above).
• ;The property \(x^a \cdot x^b = x^{a+b}\), which simplifies the multiplication of like bases.

In the context of the original problem, once you have \(r^{13}\) from the power of a product rule and \(r^{1/2}\) from the \(2r^{1/2}\) term, you use this additive property.
Add the exponents: \(r^{13} \cdot r^{1/2} = r^{13+1/2} = r^{27/2}\).
This manipulation turns multiple factors into a single term with a neat and combined exponent, making complex expressions much easier to handle. Understanding these properties allows one to deconstruct and reconstruct expressions in algebra with ease.

Simplifying Expressions

Simplifying expressions is the process of making them easier to work with while preserving their value. In this exercise, we've simplified a complex expression step by step, using the power of a product rule and properties of exponents.
Initially, the expression \((8r)^{13}(2r^{1/2})\) seems daunting. However, by leveraging the rules, we systematically simplify each part:

• ;\((8r)^{13}\) becomes \(8^{13} \cdot r^{13}\), using the power of a product.
\(r^{13} \cdot r^{1/2}\) condenses to \(r^{27/2}\), with help from the properties of exponents.

The constant multiplier \(8^{13}\) is represented by understanding its base, as follows: \(8^{13} = (2^3)^{13} = 2^{39}\) which, when multiplied by 2, becomes \(2 \cdot 2^{39} = 2^{40}\).
This thorough approach breaks down the expression into digestible parts: \(2^{40} \cdot r^{27/2}\).
Simplifying allows a clear and concise result, saving time and effort, especially with large numbers.