Problem 13 - Algebra 1: Concepts and Skills

Question

Rewrite the expression using rational exponent notation. $$ (\sqrt{16})^{5} $$

Step-by-Step Solution

Answer

\((\sqrt{16})^{5} = 16^{2.5}\)

Step 1: Identify base number and rewrite square root into exponent form

First, focus on the square root portion of the expression. Remember that \(\sqrt{a}\) can also be written as \(a^{1/2}\). You can rewrite the square root of 16 as \(16^{1/2}\). Our expression becomes: \((16^{1/2})^{5}\).

Step 2: Apply power of a power rule

The power of a power rule in mathematics states that for any real numbers a, n, and m, \((a^n)^m=a^{n*m}\). Apply this rule to rewrite the expression: \((16^{1/2*5})\).

Step 3: Simplify the exponent

In Step 2, we found that the expression becomes \(16^{1/2*5}\). When we multiply 5 by 1/2, we get 5/2 or 2.5. So our expression now is \(16^{2.5}\). Thus, the expression \((\sqrt{16})^{5}\) is equivalent to \(16^{2.5}\) in rational exponent notation.

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental concept when working with exponents. It facilitates dealing with expressions where an exponent is raised to another power. More technically, if you have an expression such as \((a^n)^m\), you can simplify it by multiplying the exponents: \(a^{n\times m}\).

This rule is particularly useful when working with more complex expressions involving exponents. It helps in quickly breaking down and simplifying expressions by reducing the number of exponentiation operations. In our example, \((16^{1/2})^5\), you apply the power of a power rule and get \(16^{1/2 \times 5} = 16^{5/2}\).

To recap the process:

• Identify the base and exponents involved.
• Multiply the exponents together.
• Write the new expression in simplified form.

Simplifying Exponents

When you're working with exponents, simplifying them can help make calculations easier and results more comprehensible. Exponents indicate how many times a number, called the base, is multiplied by itself. To simplify expressions with exponents, perform operations such as multiplication, division, or exponentiation in a structured manner.

In the problem given, we initially express the radical \(\sqrt{16}\) as \(16^{1/2}\). Next, you multiply \(1/2\) by \(5\) to simplify the exponent, reaching \(16^{5/2}\).

Here are key steps for simplifying exponents:

• Convert any radicals to an exponent form when needed.
• Apply the power rules as necessary, such as the power of a power rule.
• Carry out arithmetic operations on the exponents.
• Use rational exponents to express roots and powers concisely.

Radical Notation

Radical notation offers another way to express powers and roots, often making calculations involving roots visually straightforward and easy to comprehend. The radical sign \(\sqrt{}\) is commonly used for square roots, but it can denote any root when a different index is used.

In many math problems, you might find it beneficial to switch between radical and rational exponent notation to simplify complex expressions. For example, \(\sqrt{a}\) can be easily expressed using rational exponents as \(a^{1/2}\), which proves helpful in applying algebraic rules like the power of a power rule.

Some functions of radical notation include:

• Simplifying expressions involving roots.
• Allowing easier computation when involved in power rules.
• Providing an alternative means to express complex expressions in a simpler or more significant form.
• Seamlessly converting between radical and rational exponent notation for flexible problem-solving.