Problem 47 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ \sqrt[6]{(723)^{3}} $$

Step-by-Step Solution

Answer

The approximated value is 26.878.

Step 1: Original Expression

The problem requires us to find the sixth root of the cube of 723. In mathematical terms, this expression is written as \( \sqrt[6]{(723)^{3}} \).

Step 2: Apply Properties of Exponents

Recall that \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \). Here, \( a = 723 \), \( m = 3 \), and \( n = 6 \). Thus, the expression becomes \( 723^{\frac{3}{6}} = 723^{\frac{1}{2}} \).

Step 3: Simplify Exponent

Notice that \( 723^{\frac{1}{2}} \) is equivalent to the square root of 723, or \( \sqrt{723} \). We will calculate this value using a calculator.

Step 4: Calculator Approximation

Using a calculator, approximate \( \sqrt{723} \). Enter '723' and then apply the square root function to obtain the result, which should be approximately 26.878.

Key Concepts

Understanding Properties of Exponents

The properties of exponents are powerful rules that help us simplify and manipulate exponential expressions. Among these properties, one of the most important is the property used in this exercise: \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \), where \( a \) is the base, \( m \) is the exponent, and \( n \) is the root. This property allows us to rewrite expressions involving roots and powers as a single exponent.

• This property is useful: It simplifies the process of solving complex expressions by reducing roots to fractional exponents.
• Application: In our exercise, we swapped the sixth root for a fractional exponent \( \frac{3}{6} \), which simplifies to \( \frac{1}{2} \).
• Outcome: This simplifies the expression to finding the square root \( 723^{\frac{1}{2}} = \sqrt{723} \).

Mastering these properties aids in simplifying expressions and solving equations efficiently.

Making Accurate Calculator Approximations

When dealing with non-integer roots or powers, such as the square root of a large number like 723, calculators come in handy. They help us find approximate values quickly and accurately. Here's how to effectively use a calculator for such calculations:

• Entering values: Begin by entering the base number. For this exercise, input '723' into your calculator.
• Applying functions: Use the square root function, often represented by \( \sqrt{x} \) or a similar symbol.
• Reading results: After applying the function, the calculator will provide an approximate decimal value. Aim for three decimal places for precision.

In practice, for \( \sqrt{723} \), the calculator yields approximately 26.878. While this result is approximate, it is accurate enough for most practical purposes and homework tasks.

Exploring the Concept of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In terms of exponents, finding the square root is equivalent to raising a number to the power of one-half.

• Definition: Mathematically, the square root of a number \( x \) is represented as \( \sqrt{x} \) or equivalently, \( x^{\frac{1}{2}} \).
• Real-world relevance: Square roots are often encountered in geometry, physics, and everyday calculations, such as finding dimensions and solving equations.
• Example: For instance, finding \( \sqrt{723} \) helps us determine a number that squares back to approximately 723, highlighting its usefulness in various contexts.

Understanding the concept of square roots not only enhances problem-solving abilities but also builds a solid foundation for more complex mathematical concepts.