Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ (2 \times 8)^{\frac{1}{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses. The expression is \((2 \times 8)\). Calculate \(2 \times 8 = 16\). Now, the expression becomes \(16^{\frac{1}{4}}\).

Step 2: Convert to Rational Number

Since \(16 = 2^4\), we can use this power of 2 to further simplify the expression. Rewrite \(16^{\frac{1}{4}}\) as \((2^4)^{\frac{1}{4}}\). By the property of exponents, \((a^m)^n = a^{m \cdot n}\). Therefore, \((2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^{1}\).

Step 3: Simplify to Final Answer

Since \(2^{1} = 2\), the expression simplifies completely to the rational number \(2\).

Key Concepts

Exponents

Understanding exponents is key to exploring expressions and simplifying them. An exponent tells us how many times a base number is used as a factor. For example, in the expression \( 2^3 \), the base is \( 2 \) and the exponent is \( 3 \). This means \( 2 \) is multiplied by itself three times: \( 2 \times 2 \times 2 = 8 \). Exponents are a shorthand way of expressing repeated multiplication.There are some special cases of exponents that are good to remember:

• Any number raised to the power of \( 1 \) is just the number itself (e.g., \(a^1 = a\)).
• Any number with an exponent of \( 0 \) is equal to \( 1 \) (except when the base is \( 0 \), e.g., \( a^0 = 1 \); this is known as the zero exponent rule).

When we deal with fractions as exponents, like \( a^{\frac{1}{n}} \), it represents the \( n \)th root of \( a \). This is crucial in understanding how to simplify expressions involving fractional exponents.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its most compact form while maintaining its equivalence. This process often makes the expression easier to work with. Consider the expression \((2 \times 8)^{\frac{1}{4}}\). To simplify, we first look at the value inside the parentheses.

• Calculate the inside: \(2 \times 8 = 16\).
• Substitute back: The expression becomes \(16^{\frac{1}{4}}\).

Next, we simplify \(16^{\frac{1}{4}}\) by recognizing \(16\) as a power of \(2\) since \(16 = 2^4\). Now, you have a scenario where you can apply properties of exponents to further simplify: \((2^4)^{\frac{1}{4}}\). This directly links to the next concept, properties of exponents.

Properties of Exponents

The properties of exponents are rules that simplify the process of working with exponential expressions. These rules are very handy when you have to simplify complex expressions.One essential property is the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\). Applying this rule to our equation \((2^4)^{\frac{1}{4}}\), we multiply the exponents: \[ 4 \cdot \frac{1}{4} = 1 \]. Thus, \((2^4)^{\frac{1}{4}} = 2^{1} = 2 \).Some other important properties include:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Product: \( (ab)^n = a^n \times b^n \)

These properties allow us to break down and rebuild expressions into simpler forms, which can be a powerful tool for solving problems.