Problem 35 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ -\sqrt[4]{\frac{b^{8}}{1,296}}, b \geq 0 $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{b^{2}}{6}\).

Step 1: Simplify the expression inside the root

The given expression is \(-\sqrt[4]{\frac{b^{8}}{1,296}}\). First, let's simplify the expression inside the fourth root: \(\frac{b^{8}}{1,296}\). Since \(b^{8} = (b^{4})^2\), this can be rewritten as \(\frac{(b^{4})^2}{1,296}\).

Step 2: Evaluate the fourth root

Now, find the fourth root: \(\sqrt[4]{\frac{b^{8}}{1,296}} = \frac{b^{8/4}}{1,296^{1/4}} = \frac{b^{2}}{6}\). Here, we used the properties of exponents and roots: \(b^{8/4} = b^2\) and since \(1,296 = 6^4\), thus \(1,296^{1/4} = 6\).

Step 3: Apply the negative sign to the evaluated root

Finally, apply the negative sign outside the root: \(-\sqrt[4]{\frac{b^{8}}{1,296}} = -\frac{b^{2}}{6}\). This is the simplification of the original expression.

Key Concepts

Fourth root

Finding the fourth root of a number is quite similar to the more familiar square root, but instead of seeking a number that, when squared, gives the original, we're looking for a number that, when raised to the fourth power, yields the original number. For instance, to find \( \sqrt[4]{x} \), we ask: "What number multiplied by itself four times is equal to \( x \)?".

• It's denoted as \( x^{1/4} \) or \( \sqrt[4]{x} \).
• If you have a number like \( 64 \), since \( 64 = 4^3 \), \( \sqrt[4]{64} = 4 \).

Understanding how to find and work with fourth roots empowers you to handle more complex mathematical problems involving higher exponents. This concept extends naturally from understanding square and cube roots. As you become more comfortable with these roots, you will find greater ease in solving various algebraic expressions.

Exponent properties

Exponent properties help us manipulate and simplify expressions involving powers. These properties include how to deal with numbers that are raised to an exponent, like \( a^b \), where \( a \) is the base and \( b \) is the exponent.

• Product of Powers: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product: \( (ab)^n = a^n \times b^n \).
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \), provided \( a eq 0 \).

In the exercise, we simplified \( b^8 \) to \( (b^4)^2 \), which leverages the 'power of a power' rule, making calculations more manageable. Focusing on these learned properties will simplify various algebraic expressions and equations in your studies.

Simplifying expressions

Simplifying expressions is a fundamental part of algebra. It involves rewriting expressions in their most concise form, ensuring they are easier to evaluate and interpret. We focused on simplifying the expression inside the fourth root and the result after applying the root itself.

• Break down complex expressions into simpler components using properties of exponents and roots.
• Recognize perfect powers like \( 1,296 = 6^4 \), making it easier to simplify \( \sqrt[4]{1,296} = 6 \).
• Apply operations, such as factoring, to simplify the expression as much as possible.

Simplification makes solving algebraic problems less cumbersome and more straightforward. Practicing with different expressions will sharpen your simplification skills, allowing you to solve problems more intuitively.

Real numbers

Real numbers include all the numbers you use in everyday life, encompassing whole numbers, integers, fractions, and decimals. They can be further divided into rational and irrational numbers. In this exercise, we worked within the set of real numbers.

• Rational Numbers: Can be expressed as fractions or ratios of integers, e.g., \( \frac{7}{2}, \frac{b^2}{6} \).
• Irrational Numbers: Cannot be neatly expressed as simple fractions, e.g., \( \pi, \sqrt{2} \).
• Every real number can be placed on an infinite number line. From left to right, numbers increase in value.

Understanding and identifying real numbers are crucial for distinguishing between different types of solutions and figuring out the set of allowable answers in an equation. Being familiar with the entire spectrum of real numbers helps ensure that your solutions account for all possible scenarios.