Problem 23 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ (\sqrt[4]{y})^{-3} $$

Step-by-Step Solution

Answer

\(1/y^{3/4}\)

Step 1: Understand the Expression

The expression given is \((\sqrt[4]{y})^{-3}\), which involves both a radical and a negative exponent. Our goal is to rewrite this expression using positive exponents.

Step 2: Rewrite the Radical as an Exponent

The fourth root of \(y\) can be expressed as \(y^{1/4}\) because the nth root of a number can be rewritten as a power of the number, specifically \(x^{1/n}\). This changes the expression to \((y^{1/4})^{-3}\).

Step 3: Apply the Power Rule for Exponents

According to the power rule \((a^m)^n = a^{m\cdot n}\), we can multiply the exponents. Therefore, \((y^{1/4})^{-3}\) becomes \(y^{1/4\cdot(-3)}\).

Step 4: Simplify the Exponents

Multiply the exponents: \(1/4 \cdot (-3) = -3/4\). This simplifies our expression to \(y^{-3/4}\).

Step 5: Convert to Positive Exponents

To convert \(y^{-3/4}\) into positive exponents, recall that \(a^{-b} = 1/a^b\). Therefore, \(y^{-3/4}\) becomes \(1/y^{3/4}\).

Key Concepts

Exponents

In algebra, **exponents** are a way to describe how many times a number, called the base, is multiplied by itself. When we see an expression like \( a^b \), \( a \) is the base, and \( b \) is the exponent. Exponents can be used to simplify multiplication and to express very large or very small numbers in a compact form.
Understanding how to manipulate exponents is crucial for tackling many algebraic problems. Here are some key points about exponents:

• A positive exponent means you multiply the base by itself the specified number of times. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
• A negative exponent indicates that we need to take the reciprocal of the base raised to the absolute value of the exponent. Thus, \( a^{-b} = 1/a^b \). For example, \( 2^{-3} = 1/2^3 = 1/8 \).
• Fractional exponents, like \( a^{1/n} \), correspond to roots. For example, \( a^{1/2} \) represents the square root of \( a \).

Understanding these concepts is the first step towards mastering expressions that involve exponents.

Radicals

**Radicals** are another way to express roots of numbers, which can often be rewritten using exponents. For example, the cube root of a number \( x \) is written \( \sqrt[3]{x} \), but it can also be expressed as \( x^{1/3} \) using exponents. This conversion between radicals and exponents is crucial for simplifying complex expressions and solving equations.
Here's a quick dive into the world of radicals:

• The square root \( \sqrt{x} \) is the most common radical, equivalent to \( x^{1/2} \) using exponents.
• Higher roots, like cube roots \( \sqrt[3]{x} \), can similarly be expressed with exponents as \( x^{1/3} \).
• The general form for a radical \( \sqrt[n]{x} \) can be written as \( x^{1/n} \).

This understanding allows you to use the properties of exponents to simplify expressions involving roots.

Simplifying Expressions

**Simplifying expressions** is the process of reducing an expression to its most concise form while still maintaining its value. This can involve combining like terms, factoring, or making use of algebraic rules, such as those governing exponents and radicals.
When simplifying expressions involving exponents and radicals, some rules are particularly helpful:

• Use the power rule for exponents: \((a^m)^n = a^{m\cdot n}\). This helps in simplifying nested exponents.
• Convert negative exponents to positive exponents using the rule \(a^{-b} = 1/a^b\). For example, \( y^{-3/4} \) can be rewritten as \( 1/y^{3/4} \).
• Radicals can be rewritten as fractional exponents to apply exponent rules more easily, making complex expressions simpler to handle.

By mastering these techniques, you can simplify and solve a wide variety of algebraic expressions more efficiently.